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Convex Geometries: recent development
by
K. Adaricheva
Institute of Mathematics of SB RAS, Novosibirsk
Convex geometries are defined in combinatorics as the finite closure systems with the anti-exchange axiom. Via the lattices of closed sets they can be linked to the lattices with the unique irredundant decompositions that were studied in 40s by R.Dilworth.
In recent paper by K.Adaricheva, V.Gorbunov and V.Tumanov "Join-semidistributive lattices and convex geometries'' (to appear in Adv.Math.) we discover a close connection
of convex geometries with lattices satisfying the quasi-identity of join-semidistributivity:
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In particular, the lattice of closed sets of any finite convex geometry is join-semidistributive, and every finite join-semidistributive lattice can be embedded into the lattice of convex sets of some convex geometry. This also determines the place of the class of join-semidistributive lattices in the whole lattice hierarchy as a class that in some sense opposes to the class of modular lattices, the latter often being linked to the closure systems with the exchange-axiom.
The paper above introduces the general notion of a convex geometry as a (not necessarily finite) closure system with the anti-exchange axiom. This allows studying a wide class of closure systems that appear in different mathematical disciplines.
In the talk we will overview the recent results about convex geometries (including the author's work but not limited to it). We mention the progress that was done toward the solutions to some problems raised in the paper cited above and tell about recent studies of such key examples of convex geometries as lattices of convex subsets of partially ordered sets, lattices of suborders of partial orders, lattices of algebraic subsets of algebraic lattices and lattices of convex subsets of vector spaces.
Date received: January 31, 2003
quantum mechanical Dirac theory of the electron
by
Adewole Kayode Ajileye
Institute of Applied Mathematics
Geometric Algebra in Quantum Mechanics These papers analyze the quantum mechanical Dirac theory of the electron with respect to its geometric structure as revealed by reformulation in terms of Spacetime Algebra. The main result is that the Dirac wave function psi can be decomposed into the invariant operator form while the unit imaginary in the Dirac equation is necessarily identified with electron spin. This striking result was first derived [1] from a formulation in the book STA, which, incidentally, already showed that imaginary scalars are superfluous in the Dirac theory. Alternative derivations more directly related to the standard matrix formulation are given in [3] and an appendix to [2]. The method employed in [2] makes it transparently clear that the socalled "Fierz identities for bilinear covariants" are trivial consequences of the above invariant form for the wave function. Paper [2] provides a compact and complete formulation and analysis of local conservation laws in the one-particle Dirac theory. Comparable derivations by standard matrix and tensor methods are nearly ten times longer, as can be seen in the work of Takabayashi referenced in [2]. An analogous treatment of local conservation laws in Schroedinger's theory plays an essential role in the Bohmian interpretation of quantum mechanics. Paper [2] makes explicit the complications of extending Bohm's approach to relativistic QM.
The nonrelativistic treatment of local conservation laws including spin is given in [5] and further discussed in [6]. The main message of these papers is that standard interpretations of quantum mechanics (including Bohm's) fail to take account of the relation between spin and imaginary numbers that is inherent in Dirac theory. The necessary connections between Dirac, Pauli and Schroedinger theories are derived in [4], where inconsistencies among standard interpretations are pointed out.
Paper [3] emphasizes the point that common interpretations of Pauli and Dirac matrices as quantum mechanical operators are unjustified and ill-conceived. GA makes it absolutely clear that these matrices represent directions in space and spacetime, with no implications about spin whatsoever. Indeed, contrary to Dirac's claim and popular belief, spin is not introduced into the Dirac theory by gamma matrices but by the definition of energy-momentum operators.
Date received: January 13, 2003
Permutable rank and its linear preservers
by
Anna Alieva
Moscow State University
Let F be an arbitrary field and Mn(F) a set of n×n-matrices over F.
The theory of linear transformations on Mn(F) preserving different matrix
invariants, relations, or properties has been intensively investigated since 1897,
when Frobenius classified the bijective determinant preservers.
In 1949 Dieudonne proposed a new approach, based on the
fundumental theorem of projective geometry, and received the following
Theorem. All bijective linear transformations T on
matrix algebra Mn(F) preserving the set of all singular matrices are of
standard form: T(X) = PXQ for all matrices X or
T(X) = P(Xt)Q for all matrices X, where Xt denotes the
transposed matrix of X.
The most complete description of the results on Linear Preserver Problems
can be found in the detailed and self-contained surveys [1, 2, 3].
We investigate linear transformations T on
matrix algebra Mn(F) that preserve permutable rank, i.e.
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Date received: December 31, 2002
Dessins d'Enfants And Normalization Curve
by
Natalia Amburg
Institute of Theoretical and Experimental Physics, Moscow, Russia
In his classical work [1] A. Grothendieck put forward a theory of dessins d'enfannts. In particular, the equivalence between categories of dessins d'enfannts and Belyi pairs establishes an approach to a visualization of smooth irreducible algebraic curve over number fields. In the present work we introduce dessins d'enfannts and Belyi function defined on reducible or singular curve. We visualize the normalization curve using the method of dessins d'enfannts.
[1] Grothendiek, A: Esquisse d'un programme. London Math. Soc. Lecture Notes Series 243. Cambridge Univ. Press, (1997) 3-43
Date received: January 31, 2003
Actions of finite quantum groups on quantum polynomials
by
V. A. Artamonov
Moscow State University
A noncommutative analog of a polynomial algebra over a field k is a quantum polynomial algebra
\Lambda generated by elements
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The aim of this study is a classification of actions of finite quantum groups on Tr×An-r in terms of H-module structure on the algebra \Lambda where H is a finite dimensional pointed Hopf algebra [].
There are some examples of actions of some special pointed finite dimensional Hopf algebras on \Lambda. Namely let U be a subgroup of a finite index in Zn and [k(Zn/U)]* the dual Hopf algebra of the group algebra k(Zn/U) of the finite factorgroup Zn/U. For any f in [k(Zn/U)]* we put f o Xv = f(v+U)Xv for any monomial Xv, where v in Zn is a multi-index. Hopf algebra [k(Zn/U)]* is provided with an involutive automorphism f --> [f\tilde] where [f\tilde](v)=f(-v). So we can form a smash product [k(Zn/U)]*\sharp k<\xi> with the cyclic group <\xi> of order 2. If r=n then the action of [k(Zn/U)]* on \Lambda can be extended to an action of the smash product where \xi o Xv=X-v for all v in Zn. Suppose that qij, 1 <= i < j <= n, are independent in the multiplicative group k* of the field k and H is a pointed finite dimensional Hopf algebra H such that if r=n=2 then dimH is not divisible neither by 4 nor by 3. Let \Lambda be a left H-module algebra. Then there exists a subgroup U in Zn of a finite index and a Hopf algebra homomorphism \Psi: H --> [k(Zn/U)]*\sharp k<\xi> such that the action of H in \Lambda is a product of \Psi and the mentioned action of [k(Zn/U)]*\sharp k<\xi > on \Lambda. If \Psi is not surjective then its image is equal to [k(Zn/U)]*. It is always the case if r < n.
Date received: January 4, 2003
Countable homogeneous structures, local clones, and constraint satisfaction
by
Manuel Bodirsky
Humboldt Universitaet zu Berlin
Many interesting constraint satisfaction problems in theoretical computer science are equivalent to the question: Is there a homomorphism of a given finite relational structure to a countable homogeneous structure of the same signature? We show that the computational complexity of such problems can be characterized by the clone of polymorphisms of the countable homogeneous structure.
Date received: January 29, 2003
A Galois connection between sets of relations and sets of surjective functions
by
Ferdinand Börner
University of Potsdam
We investigate a Galois Connection Inv-sPol between sets of relations and sets of surjective functions on a finite basic set A. This connection is obtained from the Galois connection Inv-Pol by restricting the set of all functions on A to the set of all surjective functions. The investigations are motivated by complexity theoretical questions concerning the quantified constraint satisfaction problem. The results are obtained during collaboration with the coauthors of [1].
The Galois closed sets of functions can be represented by surjectively generated clones, and the Galois closed sets of relations are relational clones, closed under the additional operation of universal quantification. The set of all surjectively generated clones forms a complete and algebraic lattice SA. Similar to the lattice LA of all clones on A, this lattice is atomic and dually atomic with finitely many atoms and coatoms.
In the case |A|=2 we give a complete description of SA. For |A| > 2 we prove some results which suggest that SA is of similar complexity as LA. Following I. Rosenbergs description of the maximal clones, we can describe the dual atoms in SA.
[1] F. Börner, A. Bulatov, A. Krokhin, P. Jeavons: Quantified Constraints and Surjective Polymorphisms. Technical Report PRG-RR-02-11, Oxford Univ. Comp. Lab., 2002.
Date received: January 31, 2003
Injective Morphisms of the Machine Semigroups
by
Jānis Buls
Department of Mathematics, University of Latvia
Coauthors: Ieva Zandere
Simulation was first discussed by Hartmanis [2] more than forty years ago. This concept describes the possibility on abstract level in which one machine could be replaced by other in applications, for example, cryptography, especially, cryptanalysis of cryptographic devices. If we like to treet the machines by semigroups as it done till now and develop the theory not only as selfsufficient discipline the connections between simulation and semigroups should be considered from every point of view too. Thus we say that a transition from machines to semigroups through some representation is successful if it adequatly characterizes the simulation.
Let V=<Q, A, B, o , * > be a Mealy machine,
where
Q, A, B are
finite, nonempty sets; o : Q×A --> Q is
a function and
* : Q×A --> B is
a surjective function.
Let T(Q) denote the semigroup of all transformations on the set Q
and let
Fun(Q, B) denote the set of all maps from Q to B. On the set
S(Q, B)=T(Q)×Fun(Q, B)
define the multiplication by
(g1, p1)(g2, p2)=(g1g2, g1p2)
where g1, g2 are elements of T(Q) and p1, p2 are elements of Fun(Q, B).
Under this operation S(Q, B)
is easily seen to be a semigroup.
Let Q={ q1, q2, ... , qk}, A={a1, a2, ... , am},
B={b1, b2, ... , bn}.
Define two mappings f1 : A --> T(Q) and
f2 : A --> Fun(Q, B) as follows. For
each element ai of A
define f1(ai) as element of T(Q) and f2(ai) as element of Fun(Q, B) by
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Definition 1 [1]. Let V=<Q, A, B>,
'V=<'Q, 'A, 'B> be machines. We say that 'V
simulates V by
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We generalize the concept of similar transformation semigroups [3] to machine semigroups as follows.
Definition 2.
Let V=<Q, A, B >,
'V=<'Q, 'A, 'B > be machines.
We say that \psi: <V> --> <'V> is the s-morphism of machine semigroup <V> to <'V> if there exist maps g : Q --> 'Q, h : B --> 'B such that the diagram
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Theorem. Let V=<Q, A, B >, 'V=<'Q, 'A, 'B > be machines. If exists the injective s-homomorphism \psi: <V> --> <'V> then 'V simulates V.
References
[1] Buls, J. (1986) Ocenka dlini slova pri modelirovanii konechnix determinirovannix avtomatov. Teoreticheskie osnovi matematicheskogo obespechenija EVM, Riga: LGU im. P. Stuchki, pp.25-35 (Russian).
[2] Hartmanis J. (1961) On the State Assignment Problem for Sequential Machines I. IRE Transactions on Electronic Computers. Vol. EC-10, No.2(June), pp.157-165.
[3] Lallement G. (1979) Semigroups and Combinatorial Applications John Wlley & Sons, New York, Chichester, Brisbane, Toronto
[4] Plotkin B. I., Greenglaz I. Ja., Gvaramija A. A. (1992) Algebraic Structures in Automata and Databases Theory World Scientific, Singapore, New Jersey, London, Hong Kong.
Date received: January 29, 2003
Graph-Based Cryptographic Protocols
by
Pino Caballero-Gil
University of La Laguna. Spain
Coauthors: Candelaria Hernández-Goya and Carlos Bruno-Castañeda
In this work we propose a new methodology for the design of two-party cryptographic protocols using tools from Graph Theory. Several graph-based algorithms are presented that allow to perform over computer networks some usual actions as simple as flipping a coin or putting a message in an envelope, and as complex as signing a contract or sending a certified mail. Furthermore new proposals are introduced in order to solve new problems such as transferring information or identifying oneself, in both cases probabilistically.
Date received: January 30, 2003
Implication Algebra
by
Ivan Chajda
Palacky University Olomouc
By an implication algebra is called a groupoid satisfying the following identities:
contraction, quasi-commutativity and exchange. It was showed by J.C.Abbott in 1967 that this binary operation can be interpreted as a connective implication in the classical propositional logic (based on a Boolean algebra). Moreover, an implication algebra induces a join-semilattice with 1 where every interval [p,1] is a Boolean algebra.
Also conversely, having such a semilattice, it induces an implication algebra and this correspondence is one-to-one.
We study non-classical logics (based on orthomodular lattices or ortholattices or lattices with involutions) which rises e.g. in the logic of quantum mechanics and we assign to every of them an "implication algebra". To these algebras can be assigned a join-semilattice with 1 where every interval [p,1]is a lattice with some additional property (orthomodular or an ortholattice or a lattice with antitone involutions).
We show also conversely that each of these semilattices induces a corresponding "implication algebra". We will unify our approach to get a common theory of implication algebras.
Date received: January 16, 2003
Knowledge representation systems and skew nearlattices
by
Janis Cirulis
University of Latvia
The concept of a knowledge representation system (kr-system, for short) we deal with is a special case of that introduced in [2]. A frame is a pair (A, V), where A is a poset treated as a category, and V is a contravariant functor A --> Set respecting coproducts. Thus, V associates a set Va with every a in A and a function Vb --> Va with every pair (a, b) in V2 such that a <= b. A kr-system is a quadruple (A, V, S, \Pi), where (A, V) is a frame, S is a set, and \Pi is a cone (\pia\colon S --> Va) in Set from S to V. Elements of A and S may be thought of as attributes and states (a <= b means that a is a part of b), respectively, those of Va as possible values of a, and \pia(s) as the value of a in the state s.
A right normal skew nearlattice (rnsn-lattice, for short) was defined in [1] as an algebra (L, \/ , \odot), where (A, \odot) is a right normal band possessing the upper bound property w.r.t. its natural ordering, and \/ is the corresponding partial join operation. A relative subalgebra of L is said to be commutative, if \odot is commutative on it. L is said to be locally commutative if every principal order ideal in it is commutative. An ideal of L is a downward closed subset closed also under existing joins (hence, a relative subalgebra of L).
A commutative skew nearlattice is known as a nearlattice [3]. We shall consider here only kr-systems with the attribute set a nearlattice.
Theorem. There is a one-to-one correspondence between frames and rnsn-lattices. Furthermore, every kr-system can be presented as a triple (W, L, \models), where W is a set, L is an rnsn-lattice, and \models is a relation on W ×L such that each set {x in L\colon w \models x} is a commutative ideal of L. Moreover, a kr-system is completely determined, up to isomorphism, by its triple.
References
[1] Cirulis, J., A class of skew nearlattices. Colloq. on Semigroups (Szeged, July 17-21, 2000), Abstracts, http//at.yorku.ca/c/a/e/c/18.htm .
[2] Cirulis, J., Are there essentially incomplete knowledge representation systems? Lect. Notes Comput. Sci 2138 (2001), 94-105.
[3] Cornish, W.H., 3-permutability and quasi-commutative BCK-algebras. Math. Japon. 25 (1980), 477-496.
Date received: January 28, 2003
On an algebraic description of the category of \Sigma-coalgebras
by
Christian Dzierzon
University of Bremen, Department of Mathematics
Let \Sigma be a signature of operation symbols \sigma with arities |\sigma| bounded by a cardinal \lambda. The category CoAlg\Sigma of the corresponding polynomial functor H\Sigma:Set --> Set, with H\SigmaX=\coprod\sigma in \Sigma{\sigma}×X|\sigma| has been proven to be locally finitely presentable by [1], showing that the \Sigma-labelled trees, considered as coalgebras, form a strong generator of finitely presentable objects in CoAlg\Sigma. Thus, CoAlg\Sigma is equivalent to the category of models of an essentially algebraic theory (see e.g. [2]). We strengthen this result by showing that CoAlg\Sigma in fact is a many-sorted variety of unary algebras.
[1] J. Adámek, H.-E. Porst: On Tree Coalgebras and Coalgebra Presentations, to appear in Theoretical Computer Science.
[2] J. Adámek, J. Rosický: Locally Presentable and Accessible Categories, Cambridge University Press, Cambridge (1994).
Date received: January 30, 2003
Minimal left ideals of 2-primitive, zero symmetric nearrings
by
Wendt Gerhard
Johannes Kepler Universitaet Linz
Primitive nearrings play an important role in the structure theory of nearrings, since, similar to ring theory, a nearring is semisimple iff it is isomorphic to a subdirect product of primitive nearrings. In this talk minimal left ideals of 2-primitive, zero symmetric nearrings are studied. We show that, under some weak additional assumptions, every such minimal left ideal is in fact a planar nearring.
Date received: January 29, 2003
Clones from filters
by
Martin Goldstern
TU Wien
There is a natural map that assigns to each uniform filter F on the natural numbers N a clone C(F) on N. The unary part of C(F) is generated by the functions h(A), A in F, where h(A) is the function h(A)(n) = min (k in A: k > n).
I will investigate clones generated by this map. In a recent paper with Shelah we have constructed an ultrafilter U such that the interval of clones above C(U) is large and has no coatoms.
Date received: January 28, 2003
On externary compatible identities
by
Ewa Graczyñska
Institute of Mathematics, Technical University of Opole. Poland.
We consider well known operators N and R on varieties V of a given type by defining N(V)=Mod(N(V)), R(V)=Mod(R(V)), where N(V) denotes the set of all "normal" identities satisfied in V, R(V) denotes the set of all "regular" identities satisfied in V, cf. [5]. We deal with some other types of identities and covarieties as well.
E(V) denotes the set of all identities (of a given type) satisfied in V. We recall the notion of "regular" and "normal" variety in the following way:
Definition 1: A variety V is normal (regular) if and only if the two-element zero-semigroup (two element sup-algebra of a given type) belongs to V.
The definition is equivalnt to the well known one, invented by I. I. Melnik and J. Plonka, which say that: a variety V is normal (regular) iff E(V)=N(V), (E(V)=R(V) respectively).
Externally compatible identities in algebras were defined by J. Plonka, see W. Chromik in [4].
We recall the definition from K. Gajewska-Kurdziel and K. Mruczek, cf. [2]: Given type (2, 1) of abelian groups with two fundamental operations: the sum x+y and inverse -x.
Definition[K. Gajewska-Kurdziel, K. Mruczek] 2001/2002: The identity of type (2, 1) is externary compatible if it is one of the form: x=x, p+q=r+s, -p=-q, for some terms p, q, r, s of type (2, 1) and x is a variable.
Definition[W. Chromik, K, Halkowska] 1991: An identity p=q of type (2,2)is externary compatible if it is of the form x = x, for a variable x or p and q are terms different of a variable and both have the same outermost functional symbol.
For a given variety V, Ex(V) denotes the set of all externary compatible identities satisfied in V.
Remark 1. The trivial identity: x=y, for different variabels x and y is not externary compatible, but in groups it is equivalent to an externary compatible one, for example: (z-z) + x = (z-z)+ y, where z is a variable (or a term). Remark 2. Given any identity p = q of type (2, 1). Then in the theory G of groups of type (2, 1) it is equivalent to an externary compatible identity, for example: (z - z) + p = (z - z) + q, where z is a variable (or a term).
THEOREM. Given a variety V of lattices of type (2,2). Then the lattice L(Ex(V)) is isomorphic to the direct product of the lattice L(V) and a three-element chain.
References: [1] K. Gajewska-Kurdziel, K. Mruczek, Sets of identities satisfied in abelian groups, Demonstratio Mathematica, vol. 35, No. 3, 2002, 447-453.
[2] K. Halkowska, B. Cholewinska, R. Wiora, Externary compatible identities of Abelian groups, Acta. Univ. Wratislav. No 1890, 1997, 163-170.
[3] W. Chromik, K. Halkowska, Subvarieties of the variety defined by externary compatible identities of distributive lattices, Demonstratio Mathematica, vol. XXIV, No. 1-2, 1991, pp. 235-240.
[4] W. Chromik, Externary compatible identities of algebras, Demonstratio Mathematica 23, 1990, 344-355.
[5] Ewa Graczynska, Universal algebra via tree operads, Oficyna Wydawnicza Politechniki Opolskiej, 2000.
Date received: January 30, 2003
Rank Inequalities over Semirings and their Linear Preservers
by
Alexander Guterman
Moscow State University
This talk is based on recent joint work with LeRoy B. Beasley.
Matrix theory over semirings is an object of intensive study during the recent years since it provides the interplay between the combinatorial matrix theory and theory of linear operators. The concept of matrix rank over semirings splits into several notions. Among them there are such classical functions as factor rank, term rank, zero-term rank, row and column ranks, etc.
In the present work we investigate the behavior of these semiring rank functions under the natural operations defined on matrix algebra. We establish the serious of upper and lower bounds which are exact and the best possible. The structure of matrix varieties which arise as extremal cases in these inequalities is far from being understood. However we give the complete characterization of bijective linear transformations on matrices that leave these varieties invariant.
Date received: December 21, 2002
From triangular scheme to Mal'tsev conditions
by
Eszter K. Horváth
University of Szeged, Bolyai Institute
Coauthors: Ivan Chajda, Gábor Czédli, Paolo Lipparini
From triangular scheme to Mal'cev conditions
Motivated by Gumm's Shifting Lemma, which asserts that congruence modular varieties satisfy a nice rectangular congruence scheme, Chajda investigated a triangular scheme, which is a consequence of congruence distributivity. Congruence distributive varieties satisfy this scheme not only for arbitrary three congruences but also for one tolerance and two congruences; i.e., the analogue of Gumm's Shifting Principle is valid. The investigations went on in different directions. First, the underlying reason for congruence schemes is that certain lattice indentities are equivalent with appropriate Horn sentences, called the shift of the lattice identity , however, not every lattice identity has a shift. Secondly, while the triangular scheme does not characterize congruence distributivity, an appropriate generalization called trapezoid scheme does. The third and probably the most important direction that grew out from the topic is the question if it is possible to put tolerances in place of all the three congruences. The answer is yes. As a special case, we obtain that in a congruence modular variety, R \cap S* subset or equal (R \cap S)* holds for any two tolerances R and S. As Radeleczki and Kearnes pointed out, this can easily be turned into a much more useful property, the so-called Tolerance Intersection Property, TIP for short, of congruence modular varieties: R* \cap S* = (R \cap S)*. TIP has some applications. It is known that Tol L, the lattice of tolerances of a lattice L, has several nice properties discovered by Bandelt. Using TIP, these properties (some of them in a weaker form) can be extended to congruence distributive or congruence modular varieties, or varieties with a majority term. For example, if an algebra A has a majority term then Tol A is 0-modular, i.e., Tol A {0} contains no pentagon; the proof now is even simpler than Bandelt original one for lattices. Another application of TIP is about Mal'tsev conditions. Using TIP now we could prove that if p <= q is a lattice identitiy strong enough to imply modularity then p <= q has a Mal'tsev condition. This Mal'tsev condition is simply the conjunction of Day's condition and the Wille- Pixley's characterization of p_3 q. Here p_3 is the {, } term which we obtain from p by replacing joins by thre-fold relation product throughout. Where p <= q has previously known Mal'tsev condition then the Mal'tsev conditon extracted form p3 subset or equal q is not as good as the known one, for it contains terms with too many variables. Much better Mal'tsev condition would come from p2 subset or equal q instead of p3 subset or equal q; the latest development is that this is possible.
Date received: January 30, 2003
Construction of Semigroups with some Exotic Properties.
by
Ilya Ivanov
Moscow State University
Coauthors: Alexei Belov
There exist semigroups, groups and rings having some characteristics which are used to be considered as exotic. Namely, there exist semigroups with non-integer Gelfand-Kirillov dimension, non-nilpotent nilsemigroups and nilrings, finitely generated infinite periodic groups, and so on.
Most of these exotic objects were originally introduced either by means of infinite sets of identities or in terms of infinite sets of defining relations. So it seems to be very interesting to find finitely presented objects with similar exotic properties. An example of finitely presented associative algebra of an intermediate growth is due to V. A. Ufnarovsky, as well as the results of G. Higman, G. P. Kukin, V. Ya. Belyaev dealt with embedding of recursive presented objects (groups, associative algebras, Lie algebras) into finitely presented (we refere to [1], [2], [4], [5], [6]).
The following theorems become an object of interest in the present discussion. Theorems 1 and 2 are about the construction a finitely presented semigroups with non-integer Gelfand-Kirillov dimension and Theorem 3 is about the construction of a finitely presented semigroup G with following properties:
i) there exist a non-nilpotent ideal I=LS, where L in G;
ii) if A in G has the form A=XYYZ, there X, Y, Z in G then LA=0.
References
[1] Ufnarovsky V. A On the algebras growth. (russian) Vestnik MGU. vol 1, 1978, 4, 59-65.
[2] Ufnarovsky V. A. Combinatorical and assimptotical methods in algebra. (russian) Results of science and tech. Vol. Modern math. problems. Moscow. : VINITI, 1990, 57, 5-177.
[3] Krause G. R., Lenagan T. H. Growth of algebras and Gelfand-Kirillov dimension. London: Pitman Adv. Publ. Program, 1985, 182.
[4]Kukin G. P. The variety of all rings has Higman's property. Algebra and Analysis. Irkutsk. 1989 91-101
[5] Bokut L. A., Kukin G. P. Algoritmic and combinatorial aldebra. Math. and its appl. 255, Kluwer Academic Publishers Group, Dordrecht, 1994. xvi+384 pp
[6] Belyaev V. Ya. Imbeddability of recursively defined inverse semigroups in finitely presented semigroups. Sibirsk. Math. Journal 25 no. 2., 1984. 50-54.
Date received: January 30, 2003
On lattices of topologies of some unary algebras
by
Anna Kartashova
Volgograd State Pedagogical University
Let <A, \Omega> be an arbitrary algebra. A topology on the set A is a topology on the algebra <A, \Omega> if each operation from \Omega is continuous with respect to this topology.
The set of all topologies on the algebra <A, \Omega> forms a complete lattice where order is induced by inclusion. This lattice is called the lattice of topologies of the algebra <A, \Omega> and is denoted by Im(A).
An algebra with one unary operation is called a unar.
It is known that there is a finite algebra with two unary operations whose congruence lattice is not isomorphic to a congruence lattice of a unar (see [1, Theorem 5.6]).
We prove the similar result for the class of lattices of topologies of algebras.
Theorem. For any positive integer n there exists an algebra <A, f, g> of order 2m+1 with two unary operations such that the lattice Im(A) is distributive, consists of 4n+1 elements and Im(A) is non-isomorphic to any lattice of topologies of unars.
[1] Johnsson J., Seifert R.L. A survey of multi-unary algebra, Mimeographed seminar notes, U.C. Berkeley, 1967.
Date received: January 28, 2003
Direct products of DRl-monoids (l-groups and certain algebras)
by
Jan Kuhr
Palacky University Olomouc
Dually residuated lattice ordered monoids (DRl-monoids) constitute a generalization of lattice ordered groups (l-groups) and embrace also non-commutative extensions of algebras closely related to logic (pseudo BL-agebras, GMV-algebras etc.). We characterize the DRl-monoids that can be obtained as a direct product of an l-group and a Brouwerian algebra, a Boolean algebra, a pseudo BL-algebra, a GMV-algebra and a positive cone of an l-group, respectively.
Date received: January 30, 2003
Weakly dicomplemented lattices and double p-algebras
by
Léonard Kwuida
Intitut für Algebra, TU Dresden
Weakly dicomplemented lattices are bounded lattices equipped with two unary operations. They were introduced by R. Wille following the need to introduce a notion of "negation" of a formal concept. A model for weakly dicomplemented lattices is the class of distributive Boolean algebras, and the purpose of much of our research is to see how far they deviate from distributive double p-algebras. In [La71] Lakser gives a description of congruences of distributive p-algebras. This result is extended to distributive double p-algebras in [Ka73] by Katriñák. We are looking for a similar description for congruences of weakly dicomplemented lattices.
Date received: January 30, 2003
On n-fold ideals in BCK-algebras based on t-norme and fuzzy point
by
Celestin Lele
Dept of Mathematics, Faculty of science, University of Yaounde I, Box 812 Cameroon
The concept of fuzzy sets was first introduced by Zadeh [9] . From that time, the theory of fuzzy sets has been developed in many directions and found applications in a wide variety of fields . It application to various mathematical contexts has given rise to what is now commonly called fuzzy mathematics. Fuzzy algebras is an important branch of fuzzy mathematics, many researchers [6] have studied some algebraic structures such as fuzzy semigroups, fuzzy groups, fuzzy rings, fuzzy ideals, fuzzy modules, fuzzy vectors spaces, fuzzy category and so on. In 1991, Xi [7] applied the concept of fuzzy sets to BCI, BCK, MV-algebras . A BCI-algebra which was introduced by Iseki and Tanaka[2] is an important class of logical algebras which originated from two different ways, One of the motivation is based on the set theory and non -classical prepositional calculus, in other ways, a BCI-algebra arose from algebra of non- classical logic as Boolean algebra arose from classical logic. One of the main problem in fuzzy mathematics is how to carry out the ordinary concepts to the fuzzy case. The difficulty lies in how to pick out the rational generalization from the large number of available approaches. It is worth noting that fuzzy ideal is different from ordinary ideal in the sense that one can not say which BCK-algebra element belongs to the fuzzy ideal under consideration and which one does not.
In this paper, using the concept of fuzzy point, we develop a general theory of fuzzy ideals in BCK-algebra with respect to a t-norm T which includes many known concepts and results as its special cases.
Keywords: BCK-algebra, fuzzy point, fuzzy n-fold ideals,t-norm. REFERENCES 1.J.Meng, Y.B.Jun and H.S.Kim, Fuzzy Implicative Ideals in BCK-Algebras, Fuzzy sets and systems 89(1997)243-24 2. K. Iseki and S. Tanaka , An Introduction to the Theory of BCK -algebra, Math. Japon, 23,(1978) 1-26 3- Y. B. Jun and C-LELE , Fuzzy point BCK/BCI-algebras, The Journal of Algebra and its Application. ( accepted) 4- C.LELE, C.Wu, P. Weke, T . Mamadou and G. Edward Njock, Fuzzy Ideals and Weak Ideals in BCK-algebras Math. Japonica 4(2001) ,599-612 5. C.LELE , Imaginable Weak ideals in BCK –algebras with respect to a t-norm, ( Submitted) 6. J. N. Mordeson and D.S. Malik, Fuzzy Commutative Algebra , Creighton University, World Scientific, Singapore, December 1998 7.O.G Xi, Fuzzy BCK-algebras, Math. Japonica 36, (1991), 935-942 8.Yisheng Huang and Zhaomu, On Ideals in BCK-Algebras , Math. Japonica 50, (1999), 211-226 9. L. A. Zadeh, Fuzzy sets, Inform. And Control, 8,(1965)338-353
Date received: January 28, 2003
Compressible modules
by
Sergei Limarenko
Moscow State Univesity
A module is called compressible if it can be embedded in each of its nonzero submodules. Compressible modules were firstly introduced by Zelmanovitz (see [1]). In this work we study some basic properties of compressible modules, their inner structure and rings of endomorphisms. We describe some new objects and construction close to compressible modules. We pay special attention to those rings which are compressible as modules over themselves. The question of classification of compressible modules and rings is also under consideration. The case of graded and super modules and rings is considered in paper [2].
[1] J.Zelmanowitz, Weakly primitive rings, Comm.Algebra, 1981, v.9, N1, 23-45,
[2] Balaba I.N., Limarenko S.V., Mikhalev A.V., Zelenov S.V. Density theorems for graded rings, "Proceedeings of International Algebraic Conference, Moscow, 2000 ", J. Math. Sci., 2002, to appear, 21 p.
Date received: January 18, 2003
On Projection Algebras
by
M. Mahmoudi
Department of Mathematics, Shahid Beheshti University, Tehran 19839, Iran.
The notion of a projection space (algebra) was first introduced in 1987 by Ehrig (&, ... ) as an algebraic version of ultrametric spaces. Computer Scientists use this notion for a formal description of parallel concurrent systems. One of the main problem in this scope is the specification of infinite objects (processes) which can not be denoted by finite terms. So, they use projection algebras as a convenient mean for algebraic specification of process algebras.
A projection space (algebra) is in fact a set with an action of a monoid M=\N\infty = \N \cup {\infty}, where \N is the set of natural numbers and n < \infty, for all n in \N with the binary operation m.n=min{ m, n}, on it.
In other words, a projection algebra is a set A together with a
family (\lambdan)n in \N\infty
of unary operations \lambdan:A --> A (called projections)
such that
|
Some algebraic notions such as purity, equational compactness, tensor products, flatness, and weekly flatness have been studied for projection algebras. Here, injectivity and Baer Criterion for projection algebras is considered.
Date received: January 31, 2003
On A New Class Of Ideals In Semirings
by
P. Mukhopadhyay
Department of Mathematics, Ramakrishna Mission Residential College, Narendrapur; affeliated to --University of Calcutta
Coauthors: M.K. Sen (Dept. of Pure Mathematics, University of Calcutta), Shamik Ghosh (Dept. of Mathematics; Jadavpur University)
The concept of p-ideal in a semiring was introduced by P. Mukhopadhyay and Shamik Ghosh in A new class of ideals in semirings; (South East Asian Bull. Math. 23, (1999) pp.253-264;).
It is well-known that a ring R contains only one additive idempotent, namely the zero element. In a semiring S with additive idempotents, the set E+(S) forms an ideal of S, which is not necessarily a k-ideal. We consider the set P+(S)={x Î S : nx=(n+1)x for some n Î N} which consists of some additively periodic elements of S. Clearly, P+(R)={0} for any ring R. We note that P+(S) is an ideal of S, which is not necessarily a k-ideal but it has the following property: In a semiring S, let a Î P+(S) such that for some b Î S and some n Î N, a+nb = (n+1)b holds. Then b Î P+(S). This motivates us to define : "An ideal I of a semiring S is called a p-ideal if for some x Î S, n Î N, nx+a = (n+1)x and a Î I implies x Î I."
Clearly, in any halfring, every ideal is p-ideal. But not all p-ideals are k-ideals, as the ideal I=3Z+0\{3} is not a k-ideal, in the halfring Z+0 of all positive integers with zero. We also note that k-ideals are not p-ideals in general. Indeed, in the semiring (Z+, max, min), In= { 1, 2, 3, ¼, n } is a k-ideal for any n Î Z+ but not a p-ideal.
In the above mentioned paper, we have developed p-simple semirings and then proving the existence of maximal p-ideals which are also k-ideals, the theory of p-semifield was developed and concept of p-primitivity was investigated.
We now define a new form of regularity in a semiring, which is compatible with the concept of p-ideal, as follows: A semiring S is called p- regular if for each a Î S, there exists some b Î S such that, na + aba = (n+1) a for some n Î N. Examples are cited to justify that there exist p-regular semirings with n ¹ 1. It is shown that a semiring S is p-regular iff for every right p-ideal A and left p-ideal B in S, we have AÇB = [^AB], where [^AB] is the smallest p-ideal in S containing AB. We then obtain several chracterizations of p-regularity of a semiring in connection with p-ideals in it. Next we have defined the p-prime ideals, p-semiprime ideals and cited interesting examples of such classes and then fully p-prime semirings were introduced. Finally we have proved several results interlinking these concepts.
Date received: December 4, 2002
Local Observables in Quantum Theory
by
Michael Olubukola Oluwatukesi
Lecturer1, University of Ado Ekiti Satellite Campus,Box 74427, victoria Island, Lagos,Nig.
The Pauli theory of electrons is formulated in the language of multivector calculus. The advantages of this approach are demonstrated in an analysis of local observables. Planck's constant is shown to enter the theory only through the magnitude of the spin. Further, it is shown that, when obtained as a limiting case of the Pauli theory, the Schroedinger theory describes a particle with constant local spin. An important consequence of this result is the realization that the usual interpretations of
Date received: January 13, 2003
The Description of Characterizable Radical Differential Ideals and Construction of Characteristic Sets
by
Alexey Ovchinnikov
The Moscow State University
In this paper we consider the property of radical differential ideal to be characterizable and give the criterion to it to be characterizable. This criterion is based on the known algorithm of decomposition of radical differential ideals to characterizable components. The Ritt's problem and its restriction to our particular task are discussed. By the way, we discuss some approaches of obtaining a characteristic set of differential ideals. We also introduce and investigate a new class of definable radical differential ideals.
General approach and basic definitions are presented in [2],[3] and you can find a decomposition algorithm in [1].
[1] Hubert E., Factorization-free Decomposition Algorithms in Differential Algebra, Journal of Symbolic Computations, 2000, 29, 641-662.
[2] Kolchin E.R., Differential Algebra and Algebraic Groups, Academic Press, 1973.
[3] Ritt J.F., Differential Algebra, volume 33, New York, AMS Colloquim, 1950.
Date received: November 18, 2002
On minimal clones
by
Péter P. Pálfy
Eötvös University, Budapest, Hungary
In my survey talk at the conference celebrating Béla Csákány's 70th birthday (Szeged, July, 2002) I formulated three problems concerning minimal clones:
1. Complete the determination of minimal clones over the 4-element set.
2. Is there a minimal clone containing more than 24 majority operations?
3. Is there a minimal clone containing infinitely many binary idempotent operations?
In the present talk I will report on some progress related to these questions.
Date received: January 28, 2003
On the proper automorphisms of universal algebras
by
Alexander Pinus
Novosibirsk State Technical University
We consider automorphisms of universal algebras which are defined using the universal algerba itself. We say that an automorphism j of algerba A is termal (conditionally termal, elementary conditionally termal) if there exists a term (conditional term, elementary conditional term) t(x) of algebra A such that j(a)=t(a) for any element a from A. We say that an automorphism j of algebra A is polynomial (conditionally polynomial, elementary conditionally polynomial) if there exist a term (conditional term, elementary conditional term) t(x, y1, ... , yn) and elements b1, ... , bn from A so that j(a)=t(a, b1, ... , bn) for any element a from A. The automorphism j of algebra A is purely proper (purely conditionally proper, purely elementary conditionally proper, proper, conditionally proper, elementary conditionally proper) if j is represented as product of a purely termal (purely conditionally termal, purely elementary conditionally termal, termal, conditionally termal, elementary conditionally termal) automorphisms of algebra A. We denote the group of all purely proper (purely conditionally proper, purely elementary conditionally proper, proper, conditionally proper, elementary conditionally proper) automorphisms of algebra A as SP Aut(A) (CSP Aut(A), ECSP Aut(A), P Aut(A), CP Aut(A), ECP Aut(A)). Let Z(G) be the center of a group G. Let also Iso(A) be the semigroup of inner isomorphisms of algebra A.
Theorem 1: For any group G, for any normal subgroup G1 of the group G and for any subgroup G2 of the group G such that G2 is subset of the intersection of G1 and Z(G) there exists an universal algebra A and an isomophism h of group G on the group Aut(A) such that h(G1)=P Aut(A) and h(G2)=SP Aut(A).
Theorem 2: a). Let A be the uniformly locally finite algebra of finite signature (finite algebra of any signature). Then for any automorphism j of algebra A the following conditions are equivalent:
1). j is from CSP Aut(A).
2). All subalgebras of algebra A are j-closed and j commutes with
every function from Iso(A).
b). For any finite algebra A and for any j from Aut(A) the following conditions are equivalent:
1). j is from ECSP Aut(A).
2). All subalgebras of algebra A are j-closed and j is from Z(Aut(A)).
Theorem 3: For any uniformly locally finite algebra A of finite signature and for any j from Aut(A) the following conditions are equivalent:
1). j is from CP Aut(A).
2). There exists a finite subalgebra A1 of algebra A such that all subalgebras L of algebra A so that A1 is a subset of L are j-closed and j commutes with any h from Iso(A) such that A1 is a subset of Dom(h).
Date received: January 17, 2003
Tarski clones of operations on binary relations and their characterization
by
Reinhard Pöschel
Technische Universität Dresden, Institut für Algebra
Coauthors: Dragan Masulovic (University Novi Sad)
In the talk we give a characterization of the following three clones of operations on binary relations: the clone of primitive-positive Tarski operations, the clone of positive Tarski operations and the clone of all Tarski operations (or the classical clone). Operations from each of the three clones can be represented by special first-order formulas; to each such formula we assign a labelled multigraph and show that an operation belongs to the respective clone if and only if the suitably transformed graph of its formula does not contain a subgraph homeomorphic to K4. Finally, we address a problem of A. Tarski to characterize closed first-order formulas that can be in a suitable way represented by an identity in the language of relation algebras.
Date received: January 10, 2003
Quasivarieties of commutative binary modes
by
Anna Romanowska
Faculty of Mathematics and Information Sciences, Warsaw University of Technology, 00-661 Warsaw, Poland
Coauthors: Katarzyna Matczak (Warsaw, Poland)
Commutative binary modes are idempotent and entropic (or medial) groupoids with commutative multiplication. Their varieties were fully described by J. Jezek and T. Kepka, and their structure by A. Romanowska and J. D. H. Smith. It is well known that each cancellative commutative binary mode embeds into a commutative quasigroup mode. In this talk we will give a full description of the lattice of quasivarieties of cancellative commutative binary modes, and show that it is isomorphic to the lattice of quasivarieties of commutative quasigroup modes. We will also show how the quasivarieties of cancellative commutative binary modes determine quasivarieties of such non-cancellative groupoids.
Date received: January 31, 2003
A notion of functionally complete for first order structures
by
Temgoua Alomo Etienne Romuald
University of Yaounde1
Coauthors: Marcel Tonga
In this work, we propose a notion of functionally complete for first order structures, and find conditions for functionally complete structure to have a compatible Pixley function which is term representable on classes.
Date received: January 29, 2003
Terminal coalgebras and tree-structures
by
Christoph Schubert
University of Bremen, Department of Mathematics
Coauthors: Christian Dzierzon
It is well-known that in the category CoAlgF of F-coalgebras for a given endofunctor F:Set --> Set the terminal object can be constructed as a limit of a certain descending chain. In case of polynomial functors F=H\Sigma for bounded signatures \Sigma, this limit-object in the corresponding category CoAlg\Sigma is interpreted as the set T of all \Sigma-labelled trees with ''tree-detupling'' dynamic \theta in [1]. In the following we give a direct and more intuitive proof of this fact, and also a direct description of the unique homomorphism (A, \alpha) --> (T, \theta) for a \Sigma-coalgebra (A, \alpha).
[1] J. Adámek, V. Koubek: On the greatest fixed point of a set functor, Theoretical Computer Science 150, S. 57-75 (1995).
Date received: January 30, 2003
The graphs of involutive divisions
by
Ekaterina S. Shemyakova
153003, 12, Samoylova str, 6, Ivanovo, Russia
In this report the definition of the graph of involutive division is introduced. The graphs of classic divisions are constructed. The criterion of noetherity for the graph global involutive divisions is obtained. A series of new examples of involute divisions including Divisions of Thomas and Janet is reduced. In general, it is received the new method for studing and obtaining involutive divisions. The work was partially supported by Russian Foundatian for Basic Research project 02-01-01033.
Date received: January 28, 2003
All Input Variables Are Essential for Almost All Trees and Logical Tree Automata
by
Slavcho Shtrakov
South West University
Coauthors: Ilya Gyudzhenov
In many papers the trees were defined as terms. As in the word case there are
many results concerning automata working over trees.
The theory of essential variables was developed in the work of S.Jablonsky, A.Salomaa, K.Chimev etc.
The present paper derive some combinatorial results which show that
almost all of the logical tree automata have the maximal complexity
w.r.t. their inputs.
It is shown that k-valued logics can be interpreted
as tree languages recognized by logical automata. This means that the most of results
obtained for k-valued functions are valid for logical tree automata.
Date received: December 20, 2002
Independent subsets in concept lattices
by
Vladimir Slezak
Palacky University Olomouc, Czech Republic
To each context (G, M, I) one can assign a context of independent sets of a given cardinality. Also to any lattice L one can assign a context of independent sets. What is a connection between these associated contexts if L is a concept lattice of (G, M, I)?
Date received: January 30, 2003
Coproducts in varieties of affine modules
by
M. Stronkowski
Warsaw University of Technology, Poland
Any variety of algebras can be considered as a category with algebras as objects and homomorphisms as morphisms. Such categories have coproducts and we can ask about their structure. The answer to this question is relevant for a description of free algebras in a variety V. It is well-known that the coproduct of family of free V-algebras XiV over sets Xi is a free V-algebra over the disjoint union of the sets Xi. Thus knowledge of "small" free algebras, for instance over one or two free generators, and a good structural description of coproducts, can provide a good description of "larger" free V-algebra. This type of consideration was undertaken by Bela Csákány [1], who has shown that under certain general conditions, the coproduct of any two algebras in a given variety V coincides with their product iff V is equivalent to a variety of semimodules. In this note we are interested in varieties in which the coproduct of any two algebras A and B is isomorphic to A ×B ×2V, where 2V is a free V-algebra over two free generators. Such types of coproducts characterize varieties of affine modules. We show that a variety V has coproducts of this type iff it is equivalent to a variety of affine modules. As corollaries we obtain the known characterizations of such varieties as idempotent central Mal'cev varieties (see [3]) and as idempotent hamiltonian regular varieties (see [2]).
[1] B. Csákány, Varieties equivalent to varieties of semimodules and modules, Acta Sci. Math. 24 (1963) 157-164, in Russian.
[2] B. Csákány, Varieties of affine modules, Acta Sci. Math. 37 (1973) 3-10.
[3] A.B. Romanowska and J.D.H. Smith, Modes, World Scientific, Singapore 2002.
Date received: January 27, 2003
On the maximal subsemigroups of the finite transformation semigropup
by
K. Todorov
South-West University, Blagoevgrad, Bulgaria
Coauthors: Iliya Gyudzhenov
The full transformation \alpha of X( < ) is
isotone if i <= j ===> i\alpha <= j\alpha;
the full transformation \alpha of the set X( < ) is
increasing isotone if for every i <= j ===> i\alpha <= j\alpha& i <= i\alpha.
We give a description of the maximal subsemigroups of all
J'k. 1 <= k <= n-1-classes and of all ideals I'k of the semigroup of
all increasing isotone transformations of a finite linearly ordered set X.
The obtained results are based on previously proved propositions stating that
elements of every J'k-class, and therefore of every ideal I'k,
can be represented as products of idempotents of the same J'k-class.
Date received: December 10, 2002
A Notion of Primality for First Order Structures
by
Marcel Tonga
Dept. of Maths.; Faculty of Science (UY1) P.O Box 812 Yaounde (CAMEROON)
Coauthors: Etienne Romuald Temgoua Alomo
Given a first order structure A = (A;F;R), a congruence q of A is called a *-congruence if for any m-ary r in R and <ai,bi> in q where i=1,...m , then <a1, ...am> belongs to r iff <b1,...,bm> belongs to r.
In this work, we formulate the notions of primality and quasi-primality with respect to *-congruences, and extend the well known characterizations of primal and quasi-primal algebras to first order structures.
Date received: January 17, 2003
Hypersubstitutions in the variety of left symmetric left distributive groupoids
by
Alena Vanzurova
Dept. Alg. and Geom., Fac. Sci., Palacky University, Olomouc, CR
We investigate normal form hypersubstitutions with respect to the variety of left symmetric idempotent left distributive groupoids. The variety under consideration includes a subvariety which is mirror to the variety SIE studied before by B.Roszkowska and K.Denecke - Sr.Anworn, and some results are analogous.
The groupoid of normal form hypersubstitutions is determined, it is proven that there are no other idempotents besides those inherited from Hyp(2), and the two-element monoid of proper normal form hypersubstitutions is found.
Date received: January 31, 2003
WDVV Equations and Associative Algebras
by
Dmitry Vasiliev
Institute of Theoretical and Experimental Physics, Moscow Institute for Physics and Technology, Russia
Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations appeared in the context of topological string theories [1]. Later they were rediscovered in much large class of physical theories (see [2] and references there in).
There are 2 different geometrical interpretations of WDVV equations. First Boris Dubrovin in [3] developed a description of WDVV solutions with help of Frobenius manifolds. The main drawback of this description is the requirement of constant metric on the moduli space. In many cases this requirement is not satisfied ([2], [4]). Next description is based on the associative algebra of differential forms (which can be build in all known examples of WDVV solutions). In this talk I will discus geometric descriptions of newly developed non-trivial example of solutions to WDVV equations [4] (quasiclassical tau-function of the multi-support solutions to matrix models).
[1] E.Witten, Nucl.Phys. B340 (1990) 281; R.Dijkgraaf, H.Verline and E.Verlinde, Nucl.Phys. B352 (1991) 59.
[2] A. Gorsky, A. Mironov. FIAN-TD-30-00, ITEP-TH-64-00, Nov 2000. 134pp. In Aratyn, H. (ed.) et al.: Integrable hierarchies and modern physical theories* 33-176.
[3] Boris Dubrovin. 204pp. In Montecatini Terme 1993, Integrable systems and quantum groups* 120-348
[4] L. Chekhov, A. Marshakov, A. Mironov, D. Vasiliev. ITEP-TH-04-03, Jan 2003. 15pp. e-Print Archive: hep-th/0301071
Date received: January 31, 2003
On the scales of computability potentials of n-element unars
by
Sergey Zhurkov
Novosibirsk State Technical University
In [1] there was given the definition of the scale < CTn; <= > of computability potentials of n-element algebras. It determines what computability potential has an arbitrary n-element universal algebra. In [2]-[3] we present some results concerning the structure of these scales - a number of its atoms, coatoms, the length of the scale < CTn; <= > . Also we considered the problem of inserting the lattice into the scale and the problem of scale diagram planarity.
Let CTn1 be the set of computability potentials of n-element unar algebras. We call the partially ordered set < CTn1; <= > the scale of computability potentials of n-element unars.
We present the following results for the scale < CTn1; <= > :
Theorem 1: a). A number of coatoms of the scale < CTn1; <= > is equal to d(n)+n-1 where d(n) is a number of prime divisors of n except 1.
b). A number of atoms of the scale < CTn1; <= > is equal to D(1)+...+D(n)+T1n-1+T2n-2+...+Tn-11 where D(i) is a number of divisors of i except 1; Tij - a number of representations of i as sum of not more than j non-zero natural numbers.
Theorem 2: For n >= 3 the scale < CTn1; <= > is not up or down semilattice.
Theorem 3: For any m < n the scale < CTm1; <= > is a retract for the scale < CTn1; <= > .
Theorem 3': The scale < CTn1; <= > is a retract for the scale < CTn; <= > .
Theorem 4: For n >= 3 the scale < CTn1; <= > can't be represented as a planar graph.
Also we have found the structure of the scales < CT21; <= > and < CT31; <= > . The following statement takes place:
Statement: Any 2-element unar is conditionally rationally equivalent to one algebra from 4 pairwise conditionally rationally non-equivalent algebras B1, ..., B4. Any 3-element unar is conditionally rationally equivalent to one algebra from 14 pairwise conditionally rationally non-equivalent algebras A1, ..., A14.
[1] A.G.Pinus, S.V.Zhurkov. On the scales of computability potentials of universal algebras. // Contributions to Galois Connections, Potsdam, 2001 (in print).
[2] A.G.Pinus, S.V.Zhurkov. Some remarks on the scales of computability potentials of n-element algebras. // Algebra and Model Theory-3, Novosibirsk State Technical University, 2001, p.107-113.
[3] A.G.Pinus, S.V.Zhurkov. On the length of the scale of computability potentials of n-element algebras. // Siberian Mathematical Journal, 2002, vol.43, No.4, p.858-863.
Date received: January 17, 2003
Essential Properties of Admissible Orderings on Differential Monomials
by
Alexey Zobnin
Moscow State University
Admissible monomial orderings play an important role in the construction of Groebner bases of polynomial ideals. In the case of rings of differential polynomials the generalizations of the Groebner-bases technique also demand the specification of an admissible ranking on differential variables or monomials.
Various requirements for those rankings and orderings concerning different methods are considered in this article. We examine some links between those requirements and look at the general properties of admissible differential orderings.
In particular, we show that any admissible ordering well-orders the set of differential monomials. We prove this fact using only three natural and essential axioms, unlike the works of V. Weispfenning, where additional axioms were needed.
Date received: January 29, 2003
On the Chekhov-Fock coordinates of "dessins d'enfants".
by
Vera Zolotarskaia
Moscow State University
There are several ways to associate a complex structure to a ribbon graph (see [2], [3], [4]). In the construction of dessins d'enfants[4] a single riemann surface is associated to each graph. We call it the Grothendieck model of a ribbon graph. The goal of the present paper is to discuss one more such construction, depending on some parameters - that of Chekhov-Fock [1]. We prove that putting all parametres equal to 0, we obtain the Grothendieck model of this graph.
[1] V.V.Fock and L.O. Chekhov, "Quantim Mapping Class Group, Pentagon Relation and Geodesics", Proceedings of the Steklov Institute of Mathematics, Vol. 226, 1999, pp. 149-163.
[2] Kontsevich M.L. "Intersection theory on the moduli space of curves", functional analysis and it's applications, 1991, 25:2 pp. 50-57, in russian.
[3] Penner R.C. "The decorated Teichmuller Space of punctured surfaces", Comm. Math. Phys., 113:2 (1987), 299-340.
[4] Shabat G.B. "Combinatorial and topological methods in the theory of algebraic curves", Theses, Moscow State University, 1998, in russian.
Date received: November 27, 2002