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Институт математики им. С. Л. Соболева СО РАН, Новосибирск E-mail:sheremet@math.nsc.ru Мальцевские чтения 2009 Универсальная алгебра Конечно базируемое многообразие частичных алгебр, в котором гомоморфизмы из конечных алгебр не вычислимы М. С. Шеремет Результаты о характеризации классов подпрямо неразложимых алгебр являются красивым и классическим фрагментом теории многообразий полных алгебр, поэтому естественно искать возможность обобщения этих результатов на частичный случай.

В работе [1] П. Бурмайстер обосновал рассмотрение в случае частичных алгебр та кого варианта подпрямых разложений A Bi, где от проекций ABi требуется iI не сюръективность, а лишь эпиморфность, т. е. чтобы образы A были порождающими множествами для сомножителей.

В [2] мы установили, что в отношении алгебраических свойств класса неразложимых алгебр условие эпиморфности проекций является “хорошим”, а условие сюръективности —“плохим”. С другой стороны, нетрудно видеть, что для любого конечно базируемого многообразия частичных алгебр разрешима следующая задача: для сюръективного гомоморфизма AB, где A, выяснить, верно ли, что B ;

при этом разрешимость аналогичной задачи для эпиморфизмов A B, где A, вообще говоря, ниоткуда не следует.

И, действительно, мы указываем многообразие частичных алгебр, задаваемое конечным числом тождеств относительно равенства Клини, в котором подобная проблема неразрешима; а именно, мы предъявляем семейство An (n ) и элементы n n n fn = fA (a), gn = gA (a), hn = hA (a) (которые всегда определены и не зависят от выбора a A) со следующим свойством: не существует алгоритма, позволяющего по любому n проверить, существует ли в гомоморфизм : ABтакой, что (fn) =(gn) = (hn).

Таким образом, различные варианты обобщения понятия подпрямого разложения на случай частичных алгебр наследуют лишь свои различные части “хороших” свойств этого понятия, имеющихся для полных алгебр; в частности, сравнение этих вариантов между собой является в общем случае невозможным.

Список литературы [1] Burmeister P. Subdirect representations by epimorphisms in quasivarieties of partial algebras. Preprint Nr. 2010, TU Darmstadt, Fachbereich Mathematik, 1998.

[2] Шеремет М. С. Неразложимые алгебры в квазимногообразиях частичных алгебр. Препринт 55, ИДМИ, Новосибирск, 2001.

Институт математики им. С. Л. Соболева СО РАН, Новосибирск E-mail:sheremet@math.nsc.ru Мальцевские чтения 2009 Универсальная алгебра Semisimple Hopf algebras V. A. Artamonov Let H be a finite dimensional semisimple Hopf algebra over an algebraically closed field k. It is assumed that char k is coprime with the dimension of H. We shall assume that in each dimension d >1 there exist at most irreducible H-modules of dimension d.

Under these assumptions it is an explicit form of the counit and the antipode in H.

Suppose that there exists only one irreducible H-module of dimension > 1 and its dimension is equal to n. The order of the group G of group-like elements of the dual Hopf algebra H is a divisor of n2 [1]. If the order of G is equal to n2 then G is an Abelian group which is a direct product G = A A of two copiers of an Abelian group A. In this case there is given a classification of H up to an isomorphism in terms of faithful projective irreducible representations of G of degree n [2]. If n>2 then H is not self-dual [2].

Research is partially supported by grants RFFI 09–01–00059.

References [1] Artamonov V. A. On semisimple finite dimensional Hopf algebras. Mat. Sbornik, 198 (2007), N 9, 3–28.

[2] Artamonov V. A., Chubarov I. A., Properties of some semisimple Hopf algebras. Contemp. Math., Proc. of the International conference dedicated to 60th anniversary of I. P. Shestakov.

Department of Algebra, Faculty of Mechanics and Mathematics, Moscow State University, Moscow E-mail:artamon@mech.math.msu.su Мальцевские чтения 2009 Универсальная алгебра CSPs of bounded width and checking for the affine type A. Bulatov Bounded width is an important property of Constraint Satisfaction Problems (CSPs) that has been intensively studied for a number of years. The bounded width conjecture, recently confirmed by Barto and Kozik, states that the CSP parametrized by a relational structure A is of bounded width if and only if the corresponding algebra Alg(A) (provided it is idempotent) generates a variety omitting the unary and affine types. In this talk we consider the complexity of the problem: Given a relational structure A, decide if algebra Alg(A) generates a variety omitting the unary and affine types. It is known that if we are given the algebra Alg(A) itself, and if it is idempotent, then the problem can be solved in polynomial time. However, when input is just a relational structure, the problem is not known to be in NP or coNP. We show that if there is a uniform polynomial time algorithm solving problems of bounded width then omitting the unary and affine types can be checked in polynomial time. A stronger version of the bounded width conjecture, would provide such an algorithm.

Simon Fraser University, Vancouver (Canada) E-mail:abulatov@cs.sfu.ca Мальцевские чтения 2009 Универсальная алгебра Equational theory of nilpotent A-loops A. V. Covalschi, V. I. Ursu Algebra L =(L, ·, /, \) of (2, 2, 2) type, where identities (x · y)/y = y\(y · x) =y · (y\x) =(x/y) · y = x, x/x = y\y -hold true is called a loop. The internal substitutions Lx,y = LxLyL-1, Rx,y = RxRyRy·x, y·x Tx = RxL-1 of loop L are defined as follows:

x -Lx,y = LxLyL-1, Rx,y = RxRyRy·x, Tx = RxL-1, y·x x where x · y = yLx = xLy (x, y L).

Loop L is called an A-loop is all its internal substitutions are automorphisms [1].

In [2] it is proved that the identities of nilpotent Moufang loops have a finite basis. This paper extends this result to cover also the nilpotent A-loops.

Lemma 1. A nilpotent and finitely generated A-loop satisfies the maximality condition.

Corollary 2. A nilpotent and finitely generated A-loop is finitely represented.

Lemma 3. A nilpotent and finitely generated A-loop is residually finite.

Theorem 4. The equational theory of a nilpotent A-loop have a finite basis.

Corollary 5. The quasiequational and equational theories of any variety of nilpotent A-loops are resoluble.

References [1] Smith J. D. H., Chein O., Pflugfelder H. O. Quasigroups and loops: Theory and applications. Berlin:

HeldermannVerlag, 1990.

[2] Ursu V. I. On identities of nilpotent Moufang loops. Rev. Roumain Math. Pures Appl., 45 (2000), N. 3, 537–548.

State Pedagogical University ”Ion Creang”, Chisinu, Moldova E-mail:AlexandruCovalschi@yahoo.com Institute of Mathematics ”Simion Stoilow”, Academy of Sciences, Bucureshti, Romania; Technical University of Moldova, Chisinu, Moldova E-mail:vasile.ursu@imar.ro, vursu@mail.md Мальцевские чтения 2009 Универсальная алгебра Bounded Boolean products of pseudo MV-algebras A. Dvureenskij, M. Hyko Algebraic contruction of Boolean powers and bounded Boolean powers were investigated for orthomodular posets by Ptk ([5]), for orthoalgebras by Foulis and Ptk ([4]) and for difference posets by Dvureenskij and Pulmannov ([1]). We extend definition of bounded Boolean power for pseudo MV-algebras. We show that bounded Boolean power of a finite Boolean algebra and a pseudo MV-algebras is isomorphic to their free product. There is a topological construction of bounded Boolean powers for arbitrary universal algebras ([2], [3]). We show that the algebraic construction is dual to the topological one.

References [1] Dvureenskij A., Pulmannov S. Difference posets, effects, and quantum measurements. Inter. J. Theor.

Phys., 33 (1994), 819–850.

[2] Foster A. L. Generalized ”Boolean” theory of universal algebras. Part I: Subdirect sums and normal representation theorem. Math. Z., 58 (1953), 306–336.

[3] Foster A. L. Generalized ”Boolean” theory of universal algebras. Part II: Identities and subdirect sums in functionally complete algebras. Math. Z., 59 (1953), 191–199.

[4] Foulis D. J., Ptk P. On the tensor product of a Boolean algebra and an orthoalgebra. Czechoslovak Math. J., 45 (120) (1995), 117–126.

[5] Ptk P. Summing of Boolean algebras and logics. Demostratio Math., 19 (1986), 349–357.

Mathematical Institute, Slovak Academy of Sciences, tefnikova 49, SK-814 73 Bratislava, Slovakia E-mail: {dvurecenskij,hycko}@mat.savba.sk Мальцевские чтения 2009 Универсальная алгебра Ternary polynomials of idempotent algebras Yu. M. Movsisyan, J. Pashazadeh Let U = (U; F ) be an algebra, i.e. a nonempty set U of elements and a class F of fundamental operations consisted of U-valued functions of several variables running over U.

We called the n-ary operations e(n)(x1, x2,..., xn) =xk (k =1, 2,..., n, n =1, 2,... ) k trivial and considered the class of all polynomial operations of algebra U = (U; F ), i.e.

the smallest class of operations on the set U containing trivial operations and closed under the composition with fundamental operations of algebra. The algebra is idempotent iff the identity f(x, x,..., x) = x holds for every its polynomial operation f. K. Urbanik characterized the set of all binary polynomials of an idempotent algebra.

A similar problem on characterization of ternary polynomials of an idempotent algebra remains open.

Theorem. Suppose that an idempotent algebra has at least one binary and one ternary polynomial operation depending on every variable, and there exists an integer r >3 such that there is not any r-ary polynomial operation depending on every variable. Then the set of its ternary polynomials form a finite DeMorgan algebra with a fixed element.

We characterized these DeMorgan algebras.

Yerevan State University, Yerevan (Armenia) E-mail:yurimovsisyan@yahoo.com Мальцевские чтения 2009 Универсальная алгебра On congruence lattices of semigroups A. L. Popovich, V. B. Repnitski Recall an element a of a complete lattice L is called compact if, for any set X L that with a X, there exists a finite subset X X such that a X. A complete lattice is called algebraic if every its element is the join of some compact elements. The well-known result of G. Grtzer and E. T. Schmidt [1] states that every algebraic lattice is represented as the congruence lattice of an algebra. At the same time, this is not true if we require finiteness of similarity type of the corresponding algebra (see [2]). The main theorem of [3] states that if unit of an algebraic lattice L is compact, then L is represented as the congruence lattice of some groupoid. The following problem is open: is every distributive algebraic lattice isomorphic to the congruence lattice of a groupoid and, moreover, of a semigroup We proved the following theorem.

Theorem. Every distributive algebraic lattice whose compact elements form a lattice with unit is isomorphic to the congruence lattice of a suitable semigroup.

References [1] Grtzer G., Schmidt E. T. Characterizations of congruence lattices of abstract algebras. Acta Sci. Math.

(Szeged), 24 (1963), 34–59.

[2] Freese R., Lampe W. A., Taylor W. Congruence lattices of algebras of fixed similarity type, I. Pacific J.

Math., 82 (1979), 59–68.

[3] Lampe W. A. Congruence lattices of algebras of fixed similarity type, II. Pacific J. Math., 103 (1982), 475–508.

Ural State University, Ekaterinburg E-mail:teila@mail.ru, vladimir.repnitskii@usu.ru Мальцевские чтения 2009 Универсальная алгебра Type decompositions of an effect algebra S. Pulmannov The paper is based on the joint work with D. Foulis [7, 8]. Effect algebras (EAs) [5], play a significant role in quantum logic, are featured in the theory of partially ordered abelian groups, and generalize the classes of orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras. We study centrally orthocomplete effect algebras (COEAs), i. e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum.

For COEAs, we introduce a general notion of decomposition into types, prove that a COEA factors uniquely as a direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice. The main tool are type-determining sets (TD): a subset K of a COEA is TD iff (i) K is closed under the suprema of arbitray subfamilies of centrally orthogonal elements; (ii) intersections of elements of K with central elements belong to K. Using TD sets, it is possible to decompose a COEA into direct summands belonging to any of the above mentioned subclasses.

References [1] Carrega J. C., Chevalier G., Mayet R. Direct decompositions of orthomodular lattices. Alg. Univ., (1990), 480–496.

[2] Chang C. Algebraic analysis of many-valued logic. Trans. Amer. Math. Soc., 88 (1958), 467–490.

[3] Dvureenskij A., Pulmannov S. New Trends in Quantum Structures. Dordrecht: Kluwer, 2000.

[4] Foulis D. J., Greechie R. J., Rttimann. Filters and supports in orthoalgebras. Int. J. Theor. Phys., (1992), N. 5, 789–807.

[5] Foulis D. J., Bennett M. K. Effect algebras and unsharp quantum logics. Found. Phys., 24 (1994), N. 10, 1331–1352.

[6] Foulis D. J, Greechie R. J. Quantum logic and partially ordered abelian groups. In: Handbook of Quantum Logic and Quantum Structures, Amsterdam: Elsevier, 2007.

[7] Foulis D. J., Pulmannov S. Centrally orthocomplete effect algebras, submitted.

[8] Foulis D. J., Pulmannov S. Type decompositions of an effect algebra, submitted.

[9] Greechie R. J., Foulis D. J., Pulmannova S. The center of an effect algebra. Order, 12 (1995), 91–106.

[10] Loomis L. H. The Lattice Theoretic Background of the Dimension Theory of Operator Algebras. Memoirs of AMS, 18 (1955).

[11] Maeda S. Dimension functions on certain general lattices. J. Sci. Hiroshima Univ., A19 (1955), 211–237.

[12] Ptk P., Pulmannov S. Orthomodular Structures as Quantum Logics. Dordrecht, Boston, London:

Kluwer Academic Publ., 1991.

[13] Ramsay A. Dimension theory in complete weakly modular orthocomplemented lattices. Trans. Amer.

Math. Soc., 116 (1965), 9–31.

Mathematical Institute, Slovak Academy of Sciences, 814 73 Bratislava, Slovakia E-mail:pulmann@mat.savba.sk Мальцевские чтения 2009 Универсальная алгебра Special elements in the lattice of overcommutative semigroup varieties revisited V. Yu. Shaprynski, B. M. Vernikov A semigroup variety is called overcommutative if it contains the variety of all commutative semigroups. The lattice of all overcommutative varieties is denoted by OC. It was verified in [1] that, for an overcommutative semigroup variety V, the following are equivalent: (i) V is a distributive element of OC; (ii) V is a codistributive element of OC; (iii) V is a standard element of OC; (iv) V is a costandard element of OC; (v) V is a neutral element of OC. Besides that, a description of overcommutative varieties with the properties (i)–(v) was proposed in [1]. In actual fact, the claim that (i)–(v) are equivalent is true but the description given in [1] is false. Here we formulate a correct description.

Let m and n be positive integers with 2 m n. A sequence of positive integers m ( 1, 2,..., m) is called a partition of n into m parts if i = n and 1 2 · · · m.

i=The set of all partitions of n into m parts is denoted by n,m. Let =( 1, 2,..., m) n,m. We define numbers q(), r() and s() by the following way: q() is the number of i’s with i = 1; r() = n - q() (in other words, r() is the sum of all i’s with i > 1); s() = max{r() - q() -, 0}, where = 0 whenever n = 3, m = 2 and = (2, 1), and = 1 otherwise. If k is a non-negative integer then (k) stands for the following partition of n + k into m + k parts: (k) =( 1, 2,..., m, 1,..., 1) (in particular, k times (0) = ). If µ = (m1, m2,..., ms) r,s then Wr,s,µ is the set of all words u with the following properties: the length of u equals r, u depends on variables x1, x2,..., xs, and the number of occurrences of xi in u equals mi for all i = 1, 2,..., s. For a partition =( 1, 2,..., m) n,m, we denote by S the semigroup variety given by all identities of the form u = v such that u, v Wn+i,m+i, for some 0 i s().

(i) Theorem. An overcommutative semigroup variety V has an arbitrary of the (equivalent) properties (i)–(v) if and only if either V coincides with the variety of all semigroups or k V = S for some set of partitions {1, 2,..., k}.

i i=References [1] Vernikov B. M. Special elements of the lattice of overcommutative semigroup varieties. Matem. Zametki, 70 (2001), 670–678 [in Russian; Engl. translation: Math. Notes, 70 (2001), 608–615].

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