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Journal articles on the topic 'Variety of algebra'

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Published: 9 March 2023

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1

GIVANT, STEVEN, and HAJNAL ANDRÉKA. "THE VARIETY OF COSET RELATION ALGEBRAS." Journal of Symbolic Logic 83, no. 04 (December 2018): 1595–609. http://dx.doi.org/10.1017/jsl.2018.48.

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AbstractGivant [6] generalized the notion of an atomic pair-dense relation algebra from Maddux [13] by defining the notion of a measurable relation algebra, that is to say, a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). In Andréka--Givant [2], a large class of examples of such algebras is constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to “shift” the operation of relative multiplication. In Givant--Andréka [8], it is shown that the class of these full coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic and complete measurable relation algebra is isomorphic to a full coset relation algebra.Call an algebra $\mathfrak{A}$ a coset relation algebra if $\mathfrak{A}$ is embeddable into some full coset relation algebra. In the present article, it is shown that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but that no finite set of sentences suffices to axiomatize the class (that is to say, the class is not finitely axiomatizable).

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2

Padmanabhan, R., and P. Penner. "A Universal Variety of Point Algebras." Algebra Colloquium 17, no. 04 (December 2010): 647–58. http://dx.doi.org/10.1142/s1005386710000623.

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Point algebras introduced by Evans are algebraic systems which capture the essence of multiplications (a,b) · (c,d)=(p,q) defined on the set of all ordered pairs of elements of a set S, where p and q are selected from among a,b,c,d by some well-defined rule. In 1961, Jonsson and Tarski gave an interesting example of a variety of algebras of type 〈2,1,1〉 for illustrating the failure of certain free algebra properties. In this paper, we show that this equational class of algebras, called the JT-variety, is a universal variety of point algebras in the sense that every variety generated by a point algebra is a reduct of the JT-variety.

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3

Dziobiak, Wieslaw. "The subvariety lattice of the variety of distributive double p-algebras." Bulletin of the Australian Mathematical Society 31, no. 3 (June 1985): 377–87. http://dx.doi.org/10.1017/s0004972700009345.

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Let L denote the subvariety lattice of the variety of distributive double p-algebras, that is, the lattice whose universe consists of all varieties of distributive double p-algebras and whose ordering is the inclusion relation. We prove in this paper that each proper filter in L is uncountable. Moreover, we prove that except for the trivial variety (the zero in L) and the variety of Boolean algebras (the unique atom in L) every other element of L, generated by a finite algebra, has infinitely many covers in L, among which at least one is not generated by any finite algebra. The former result strengthens a result of Urquhart who showed that the lattice L is uncountable. On the other hand, both of our results indicate a high complexity of the lattice L at least in comparison with the subvariety lattice of the variety of distributive p-algebras, since a result of Lee shows that the latter lattice forms a chain of type ω + 1 and every cover in it of the variety generated by a finite algebra is itself generated by a finite algebra.

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4

Benanti, Francesca, Onofrio M. Di Vincenzo, and Vincenzo Nardozza. "*-Subvarieties of the Variety Generated by." Canadian Journal of Mathematics 55, no. 1 (February 1, 2003): 42–63. http://dx.doi.org/10.4153/cjm-2003-002-7.

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AbstractLet be a field of characteristic zero, and * = t the transpose involution for the matrix algebra M2(). Let be a proper subvariety of the variety of algebras with involution generated by . We define two sequences of algebras with involution Rp, Sq, where p, q ∊ . Then we show that T*() and T*(Rp ⊕ Sq) are *-asymptotically equivalent for suitable p, q.

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5

Remm, Elisabeth. "3-Dimensional Skew-symmetric Algebras and the Variety of Hom-Lie Algebras." Algebra Colloquium 25, no. 04 (December 2018): 547–66. http://dx.doi.org/10.1142/s100538671800038x.

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An algebra is called skew-symmetric if its multiplication operation is a skewsymmetric bilinear application. We determine all these algebras in dimension 3 over a field of characteristic different from 2. As an application, we determine the subvariety of 3-dimensional Hom-Lie algebras. For this type of algebra, we study also the case of dimension 4.

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6

Chen, Cong. "The nilpotent variety of W(1;n)p is irreducible." Journal of Algebra and Its Applications 18, no. 03 (March 2019): 1950056. http://dx.doi.org/10.1142/s0219498819500567.

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In the late 1980s, Premet conjectured that the nilpotent variety of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic [Formula: see text] is irreducible. This conjecture remains open, but it is known to hold for a large class of simple restricted Lie algebras, e.g. for Lie algebras of connected algebraic groups, and for Cartan series [Formula: see text] and [Formula: see text]. In this paper, with the assumption that [Formula: see text], we confirm this conjecture for the minimal [Formula: see text]-envelope [Formula: see text] of the Zassenhaus algebra [Formula: see text] for all [Formula: see text].

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7

KHARLAMPOVICH, O., and D. GILDENHUYS. "THE WORD PROBLEM FOR SOME VARIETIES OF SOLVABLE LIE ALGEBRAS." International Journal of Algebra and Computation 04, no. 03 (September 1994): 481–91. http://dx.doi.org/10.1142/s0218196794000117.

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The word problem is said to be solvable in a variety of Lie algebras if it is solvable in every algebra, finitely presented in this variety. Let [Formula: see text] denote the variety of (2-step nilpotent)-by-abelian Lie algebra and [Formula: see text] the variety of abelian-by-(2-step nilpotent) Lie algebras. It is proved that the word problem is unsolvable in the “interval” of varieties containing the variety [Formula: see text] (of centre-by-[Formula: see text] Lie algebras over a field of characteristic zero), and contained in the variety [Formula: see text].

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8

MIKHALEV, ALEXANDER A., and JIE-TAI YU. "STABLE EQUIVALENCE PROBLEMS FOR FREE ALGEBRAS WITH THE NIELSEN-SCHREIER PROPERTY." International Journal of Algebra and Computation 11, no. 06 (December 2001): 779–86. http://dx.doi.org/10.1142/s0218196701000747.

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A variety of algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras. For free algebras of finite ranks of Schreier varieties we prove that if two systems of elements are stably equivalent, then they are equivalent. We define the rank of an endomorphism of a free algebra of a Schreier variety and prove that an injective endomorphism of maximal rank does not change the rank of elements of maximal rank.

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9

Abad, M., and J. P. Díaz Varela. "Free algebras in the variety of three-valued closure algebras." Journal of the Australian Mathematical Society 72, no. 2 (April 2002): 181–98. http://dx.doi.org/10.1017/s1446788700003839.

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AbstractIn this paper, the variety of three-valued closure algebras, that is, closure algebras with the property that the open elements from a three-valued Heyting algebra, is investigated. Particularly, the structure of the finitely generated free objects in this variety is determined.

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10

BLOOM, STEPHEN L., and ZOLTÁN ÉSIK. "Varieties generated by languages with poset operations." Mathematical Structures in Computer Science 7, no. 6 (December 1997): 701–13. http://dx.doi.org/10.1017/s0960129597002442.

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A V-labelled poset P can induce an operation on the languages on any fixed alphabet, as well as an operation on labelled posets (as noticed by Pratt and Gischer (Pratt 1986; Gischer 1988)). For any collection X of V-labelled posets and any alphabet Σ we obtain an X-algebra ΣX of languages on Σ. We consider the variety Lang(X) generated by these algebras when X is a collection of nonempty ‘traceable posets’. The current paper contains several observations about this variety. First, we use one of the basic results in Bloom and Ésik (1996) to show that a concrete description of the A-generated free algebra in Lang(X) is the X-subalgebra generated by the singletons (labelled a∈A) in the X-algebra of all A-labelled posets. Equipped with an appropriate ordering, these same algebras are the free ordered algebras in the variety Lang(X)[les ] of ordered language X-algebras. Further, if one enriches the language algebras by adding either a binary or infinitary union operation, the free algebras in the resulting variety are described by certain ‘closed’ subsets of the original free algebras. Second, we show that for ‘reasonable sets’ X, the variety Lang(X) has the property that for each n[ges ]2, the n-generated free algebra is a subalgebra of the 1-generated free algebra. Third, knowing the free algebras enables us to show that these varieties are generated by the finite languages on a two-letter alphabet.

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11

BENEŠ, THOMAS, and DIETRICH BURDE. "CLASSIFICATION OF ORBIT CLOSURES IN THE VARIETY OF THREE-DIMENSIONAL NOVIKOV ALGEBRAS." Journal of Algebra and Its Applications 13, no. 02 (October 10, 2013): 1350081. http://dx.doi.org/10.1142/s0219498813500813.

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We classify the orbit closures in the variety [Formula: see text] of complex, three-dimensional Novikov algebras and obtain the Hasse diagrams for the closure ordering of the orbits. We provide invariants which are easy to compute and which enable us to decide whether or not one Novikov algebra degenerates to another Novikov algebra.

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12

Kaygorodov, Ivan, and Yury Volkov. "The Variety of Two-dimensional Algebras Over an Algebraically Closed Field." Canadian Journal of Mathematics 71, no. 4 (October 16, 2018): 819–42. http://dx.doi.org/10.4153/s0008414x18000056.

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AbstractThe work is devoted to the variety of two-dimensional algebras over algebraically closed fields. First we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of certain algebra series in the variety of two-dimensional algebras. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.

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13

Adams, M. E. "Principal congruences in de Morgan algebras." Proceedings of the Edinburgh Mathematical Society 30, no. 3 (October 1987): 415–21. http://dx.doi.org/10.1017/s0013091500026808.

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A congruence relation θ on an algebra L is principal if there exist a, b)∈L such that θ is the smallest congruence relation for which (a, b)∈θ. The property that, for every algebra in a variety, the intersection of two principal congruences is again a principal congruence is one that is known to be shared by many varieties (see, for example, K. A. Baker [1]). One such example is the variety of Boolean algebras. De Morgan algebras are a generalization of Boolean algebras and it is the intersection of principal congruences in the variety of de Morgan algebras that is to be considered in this note.

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14

Anantpiniwatna, Apinant, and Tiang Poomsa-Ard. "IDENTITIES IN BIREGULAR LEFTMOST GRAPH VARIETIES OF TYPE (2,0)." Asian-European Journal of Mathematics 02, no. 01 (March 2009): 1–17. http://dx.doi.org/10.1142/s1793557109000029.

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Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies a term equation s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A class of graph algebras V is called a graph variety if V = ModgΣ where Σ is a subset of T(X) × T(X). A graph variety V' = ModgΣ' is called a biregular leftmost graph variety if Σ' is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity in a variety V if G satisfies s ≈ t for all G ∈ V. In this paper we characterize identities in each biregular leftmost graph variety. For identities, varieties and other basic concepts of universal algebra see e.g. [1].

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15

Yao, Yu-Feng, and Hao Chang. "Commuting variety of Witt algebra." Frontiers of Mathematics in China 13, no. 5 (September 14, 2018): 1179–87. http://dx.doi.org/10.1007/s11464-018-0725-9.

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16

McKENZIE, RALPH, and JAPHETH WOOD. "THE TYPE SET OF A VARIETY IS NOT COMPUTABLE." International Journal of Algebra and Computation 11, no. 01 (February 2001): 89–130. http://dx.doi.org/10.1142/s0218196701000504.

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Several problems involving the local structure of finite algebras are shown to be unsolvable by interpreting the halting problem of Turing machines. Specifically, these problems are to decide, given a finite algebra A, whether the type-set of the variety generated by A, which is a subset of {1,2,3,4,5}, omits 2, or 3, or 4, or 5, respectively.

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17

McNULTY, GEORGE F., ZOLTÁN SZÉKELY, and ROSS WILLARD. "EQUATIONAL COMPLEXITY OF THE FINITE ALGEBRA MEMBERSHIP PROBLEM." International Journal of Algebra and Computation 18, no. 08 (December 2008): 1283–319. http://dx.doi.org/10.1142/s0218196708004913.

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We associate to each variety of algebras of finite signature a function on the positive integers called the equational complexity of the variety. This function is a measure of how much of the equational theory of a variety must be tested to determine whether a finite algebra belongs to the variety. We provide general methods for giving upper and lower bounds on the growth of equational complexity functions and provide examples using algebras created from graphs and from finite automata. We also show that finite algebras which are inherently nonfinitely based via the shift automorphism method cannot be used to settle an old problem of Eilenberg and Schützenberger.

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18

IM, BOKHEE, and JONATHAN D. H. SMITH. "REPRESENTATION THEORY FOR VARIETIES OF COMTRANS ALGEBRAS AND LIE TRIPLE SYSTEMS." International Journal of Algebra and Computation 21, no. 03 (May 2011): 459–72. http://dx.doi.org/10.1142/s0218196711006315.

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For a variety of comtrans algebras over a commutative ring, representations of algebras in the variety are identified as modules over an enveloping algebra. In particular, a new, simpler approach to representations of Lie triple systems is provided.

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19

Massé, C., H. Wang, and S. L. Wismath. "Minimal Characteristic Algebras for Leftmost k-Normal Identities." Algebra Colloquium 17, no. 01 (March 2010): 27–42. http://dx.doi.org/10.1142/s1005386710000052.

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A characteristic algebra for a hereditary property of identities is an algebra [Formula: see text] which generates the variety determined by all identities with that property. We use Płonka's construction, and known minimal characteristic algebras for the k-normality and leftmost properties, to construct minimal characteristic algebras of type (2) for leftmost k-normality for 1 ≤ k ≤ 3, and show that Płonka's construction does not always give a minimal algebra.

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20

Miller, John Boris. "Strictly real Banach algebras." Bulletin of the Australian Mathematical Society 47, no. 3 (June 1993): 505–19. http://dx.doi.org/10.1017/s000497270001532x.

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A complex Banach algebra is a complexification of a real Banach algebra if and only if it carries a conjugation operator. We prove a uniqueness theorem concerning strictly real selfconjugate subalgebras of a given complex algebra. An example is given of a complex Banach algebra carrying two distinct but commuting conjugations, whose selfconjugate subalgebras are both strictly real. The class of strictly real Banach algebras is shown to be a variety, and the manner of their generation by suitable elements is proved. A corollary describes some strictly real subalgebras in Hermitian Banach star algebras, including C* algebras.

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21

KEARNES, KEITH A., and EMIL W. KISS. "RESIDUAL SMALLNESS AND WEAK CENTRALITY." International Journal of Algebra and Computation 13, no. 01 (February 2003): 35–59. http://dx.doi.org/10.1142/s0218196703001237.

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We develop a method of creating skew congruences on subpowers of finite algebras using groups of twin polynomials, and apply it to the investigation of residually small varieties generated by nilpotent algebras. We prove that a residually small variety generated by a finite nilpotent (in particular, a solvable E-minimal) algebra is weakly abelian. Conversely, we show in two special cases that a weakly abelian variety is residually bounded by a finite number: when it is generated by an E-minimal, or by a finite strongly nilpotent algebra. This establishes the RS-conjecture for E-minimal algebras.

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Manyuen, C., P. Jampachon, and T. Poomsa-ard. "Graph variety generated by linear terms." Asian-European Journal of Mathematics 12, no. 05 (September 3, 2019): 1950074. http://dx.doi.org/10.1142/s1793557119500748.

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Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type [Formula: see text]. We say that a graph [Formula: see text] satisfies a term equation [Formula: see text] if the corresponding graph algebra [Formula: see text] satisfies [Formula: see text]. The set of all term equations [Formula: see text], which the graph [Formula: see text] satisfies, is denoted by [Formula: see text]. The class of all graph algebras satisfy all term equations in [Formula: see text] is called the graph variety generated by [Formula: see text] denoted by [Formula: see text]. A term is called a linear term if each variable which occurs in the term, occurs only once. A term equation [Formula: see text] is called a linear term equation if [Formula: see text] and [Formula: see text] are linear terms. This paper is devoted to a thorough investigation of graph varieties defined by linear term equations. In particular, we give a complete description of rooted graphs generating a graph variety described by linear term equations.

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23

Marczak, Adam W., and Jerzy Płonka. "Mapping Extension of an Algebra." Algebra Colloquium 16, no. 03 (September 2009): 479–94. http://dx.doi.org/10.1142/s1005386709000455.

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A new construction of algebras called a mapping extension of an algebra is here introduced. The construction yields a generalization of some classical constructions such as the nilpotent extension of an algebra, inflation of a semigroup but also the square extension construction introduced recently for idempotent groupoids. The mapping extension construction is defined for algebras of any fixed type, however nullary operation symbols are here not admitted. It is based on the notion of a retraction and some system of mappings. A mapping extension of a given algebra is constructed as a counterimage algebra by a specially defined retraction. Varieties of algebras satisfying an identity φ(x) ≈ x for a term φ not being a variable (such as varieties of lattices, Boolean algebras, groups and rings) are especially interesting because for such a variety [Formula: see text], all mapping extensions by φ of algebras from [Formula: see text] form an equational class. In the last section, combinatorial properties of the mapping extension construction are considered.

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24

Dekov, Deko V. "Embeddability and the word problem." Journal of Symbolic Logic 60, no. 4 (December 1995): 1194–98. http://dx.doi.org/10.2307/2275882.

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Let be a finitely presented variety with operations Ω and let be the variety having the same set of operations Ω but defined by the empty set of identities. A partial-algebra is a set P with a set of mappings containing for each n-ary operation f of Ω a mapping , where D ⊆ Pn. An incomplete -algebra is a partial -algebra which satisfies the defining identities of , insofar as they can be applied to the partial operations of (Trevor Evans [4, p. 67]). We call an incomplete -algebra a partial Evans-algebra if it can be embedded in a member of the variety .If the class of all partial Evans -algebras is (first-order) finitely axiomatizable, then the word problem for the variety is solvable. (Evans [4, 5]). In 1953 Evans [5, p. 79] raised the question of whether the converse is true. In this paper we show that the answer is in the negative.Let CSg denote the variety of commutative semigroups. We call an incomplete CSg-algebra an incomplete commutative semigroup and we call a partial Evans CSg-algebra a partial Evans commutative semigroup. It is known (A. I. Malcev [9] see also Evans [6]) that the variety of commutative semigroups has solvable word problem. We show (Theorem 1) that the class of all partial Evans commutative semigroups is not finitely axiomatizable. Therefore the solvability of the word problem for the variety of commutative semigroups does not imply finite axiomatizability of the class of all partial Evans commutative semigroups.

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25

BLOOM, STEPHEN L., and ZOLTÁN ÉSIK. "ITERATION ALGEBRAS." International Journal of Foundations of Computer Science 03, no. 03 (September 1992): 245–302. http://dx.doi.org/10.1142/s0129054192000164.

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We assume the reader has some familiarity with theories and iteration theories. The main topic of the paper is properties of varieties of iteration algebras. After a preliminary section which contains all of the necessary definitions, we spend some time on a coproduct construction which is needed to prove a fundamental lemma: for each iteration theory T, any T-algebra is a retract of a T-iteration algebra. The proof of the lemma shows that only one property of iteration theories is used: the parameter identity. Hence, the lemma applies to any preiteration theory in which this identity is valid. It follows from this fact that the variety of all T-iteration algebras has “nice” properties only when every T-algebra is an iteration algebra. Some of the possible pathology in varieties of iteration algebras is demonstrated. It is shown that for each set Z of non-negative integers there is a variety of iteration algebras having an n-generated free algebra iff n∈Z. Also given is a theorem characterizing certain functors between varieties of iteration theories which are induced by iteration theory morphisms. We find an explicit description of all of the theory congruences on theories of partial functions. The last section is connected with the strong iteration algebras introduced in a paper by Ésik (1983).

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Polovinkina, A. V., and T. V. Skoraya. "CONDITION OF FINITENESS OF COLENGTH OF VARIETY OF LEIBNITZ ALGEBRAS." Vestnik of Samara University. Natural Science Series 20, no. 10 (May 29, 2017): 84–90. http://dx.doi.org/10.18287/2541-7525-2014-20-10-84-90.

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This paper is devoted to the varieties of Leibnitz algebras over a ﬁeld of zero characteristic. All information about the variety in case of zero characteristic of the base ﬁeld is contained in the space of multilinear elements of its relatively free algebra. Multilinear component of variety is considered as a module of symmetric group and splits into a direct sum of irreducible submodules, the sum of multiplicities of which is called colength of variety. This paper investigates the identities that are performed in varieties with ﬁnite colength and also the relationship of this varieties with known varieties of Lie and Leibnitz algebras with this property. We prove necessary and suﬃcient condition for a ﬁniteness of colength of variety of Leibnitz algebras.

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Sun, Xiaosong. "Automorphisms of the endomorphism semigroup of a free algebra." International Journal of Algebra and Computation 25, no. 08 (December 2015): 1223–38. http://dx.doi.org/10.1142/s0218196715500381.

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We suggest a new method for describing automorphisms of the endomorphism semigroup of a free algebra in a variety of algebras. As an application, we describe automorphisms of the endomorphism semigroup of a free Poisson algebra over an arbitrary field [Formula: see text].

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Denecke, K., J. Koppitz, and R. Marszałek. "Derived Varieties and Derived Equational Theories." International Journal of Algebra and Computation 08, no. 02 (April 1998): 153–69. http://dx.doi.org/10.1142/s0218196798000090.

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This paper describes a derivation process for varieties and equational theories using the theory of hypersubstitutions and M-hyperidentities. A hypersubstitution σ of type τ is a map which takes each n-ary operation symbol of the type to an n-ary term of this type. If [Formula: see text] is an algebra of type τ then the algebra [Formula: see text] is called a derived algebra of [Formula: see text]. If V is a class of algebras of type τ then one can consider the variety vσ(V) generated by the class of all derived algebras from V. In the first two sections the necessary definitions are given. In Sec. 3 the properties of derived varieties and derived equational theories are described. On the set of all derived varieties of a given variety, a quasiorder is developed which gives a derivation diagram. In the final section the derivation diagram for the largest solid variety of medial semigroups is worked out.

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DI NOLA, A., P. FLONDOR, and B. GERLA. "COMPOSITION ON MV-ALGEBRAS." Journal of Algebra and Its Applications 05, no. 04 (August 2006): 417–39. http://dx.doi.org/10.1142/s0219498806001818.

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In this paper we introduce an extension of MV-algebras obtained by adding a binary operation and a constant, with the aim of modelling composition of functions. The variety of Composition MV-algebra (CMV-algebra, for short) is defined and some results regarding ideals and congruences are stated. Further, we define modules over CMV-algebras showing that to any substitution corresponds an endomorphism of modules.

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NURAKUNOV, ANVAR M., and MICHAŁ M. STRONKOWSKI. "PROFINITENESS IN FINITELY GENERATED VARIETIES IS UNDECIDABLE." Journal of Symbolic Logic 83, no. 04 (December 2018): 1566–78. http://dx.doi.org/10.1017/jsl.2017.89.

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AbstractProfinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety ${\cal V}$ is standard if every Boolean topological algebra with the algebraic reduct in ${\cal V}$ is profinite.We show that there is no algorithm which takes as input a finite algebra A of a finite type and decide whether the variety $V\left( {\bf{A}} \right)$ generated by A is standard. We also show the undecidability of some related properties. In particular, we solve a problem posed by Clark, Davey, Freese, and Jackson.We accomplish this by combining two results. The first one is Moore’s theorem saying that there is no algorithm which takes as input a finite algebra A of a finite type and decides whether $V\left( {\bf{A}} \right)$ has definable principal subcongruences. The second is our result saying that possessing definable principal subcongruences yields possessing finitely determined syntactic congruences for varieties. The latter property is known to yield standardness.

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31

Chirivì, Rocco. "On some properties of LS algebras." Communications in Contemporary Mathematics 22, no. 02 (December 3, 2018): 1850085. http://dx.doi.org/10.1142/s0219199718500852.

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The discrete LS algebra over a totally ordered set is the homogeneous coordinate ring of an irreducible projective (normal) toric variety. We prove that this algebra is the ring of invariants of a finite abelian group containing no pseudo-reflection acting on a polynomial ring. This is used to study the Gorenstein property for LS algebras. Further we show that any LS algebra is Koszul.

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Zhitomirski, G. "Types of points and algebras." International Journal of Algebra and Computation 28, no. 08 (December 2018): 1717–30. http://dx.doi.org/10.1142/s0218196718400167.

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The connection between classical model theoretical types (MT-types) and logically-geometrical types (LG-types) introduced by B. Plotkin is considered. It is proved that MT-types of two [Formula: see text]-tuples in two universal algebras coincide if and only if their LG-types coincide. Two problems set by B. Plotkin are considered: (1) let two tuples in an algebra have the same type, does it imply that they are connected by an automorphism of this algebra? and (2) let two algebras have the same type, does it imply that they are isomorphic? Some varieties of universal algebras are considered having in view these problems. In particular, it is proved that if a variety is hopfian or co-hopfian, then finitely generated free algebras of such a variety are completely determined by their type.

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33

Mishchenko, S. P., and O. V. Shulezhko. "On almost nilpotent varieties in theclass of commutative metabelian algebras." Vestnik of Samara University. Natural Science Series 21, no. 3 (May 19, 2017): 21–28. http://dx.doi.org/10.18287/2541-7525-2015-21-3-21-28.

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A well founded way of researching the linear algebra is the study of it using the identities, consequences of which is the identity of nilpotent. We know the Nagata-Higman’s theorem that says that associative algebra with nil condition of limited index over a ﬁeld of zero characteristic is nilpotent. It is well known the result of E.I.Zel’manov about nilpotent algebra with Engel identity. A set of linear algebras where a ﬁxed set of identities takes place, following A.I. Maltsev, is called a variety. The variety is called almost nilpotent if it is not nilpotent, but each its own subvariety is nilpotent. Here in the case of the main ﬁeld with zero characteristic, we proved that for any positive integer m there exist commutative metabelian almost nilpotent variety of exponent is equal to m.

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BREMNER, MURRAY R., and IRVIN R. HENTZEL. "IDENTITIES RELATING THE JORDAN PRODUCT AND THE ASSOCIATOR IN THE FREE NONASSOCIATIVE ALGEBRA." Journal of Algebra and Its Applications 05, no. 01 (February 2006): 77–88. http://dx.doi.org/10.1142/s0219498806001594.

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We determine the identities of degree ≤ 6 satisfied by the symmetric (Jordan) product a○b = ab + ba and the associator [a,b,c] = (ab)c - a(bc) in every nonassociative algebra. In addition to the commutative identity a○b = b○a we obtain one new identity in degree 4 and another new identity in degree 5. We demonstrate the existence of further new identities in degree 6. These identities define a variety of binary-ternary algebras which generalizes the variety of Jordan algebras in the same way that Akivis algebras generalize Lie algebras.

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35

Skoraya, T. V., and Yu Yu Frolova. "ON SEVERAL VARIETIES OF LEIBNIZ ALGEBRAS." Vestnik of Samara University. Natural Science Series 17, no. 5 (June 14, 2017): 71–80. http://dx.doi.org/10.18287/2541-7525-2011-17-5-71-80.

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The paper is devoted to two new results concerning varieties of Leibniz algebras. In case of prime characteristic p we construct an example of a non-nilpotent variety of Leibniz algebras with Engel condition. In case of field of characteristic zero we obtain a new result concerning the space of multilinear componentsof the variety of left-nilpotent Leibniz algebra of class three.

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KUN, GÁBOR, and VERA VÉRTESI. "THE MEMBERSHIP PROBLEM IN FINITE FLAT HYPERGRAPH ALGEBRAS." International Journal of Algebra and Computation 17, no. 03 (May 2007): 449–59. http://dx.doi.org/10.1142/s0218196707003640.

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The membership problem asks whether a finite algebra belongs to the variety generated by another finite algebra. In some sense the β-function is the measure of the complexity of the membership problem. We investigate the β-function for finite flat hypergraph algebras and prove that in general it is not bounded by any polynomial.

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37

SARAMAGO, M. J., and H. A. PRIESTLEY. "OPTIMAL NATURAL DUALITIES: THE STRUCTURE OF FAILSETS." International Journal of Algebra and Computation 12, no. 03 (June 2002): 407–36. http://dx.doi.org/10.1142/s0218196702000791.

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B. A. Davey and H. A. Priestley have investigated the optimality of dualities on a quasivariety [Formula: see text], where [Formula: see text] is a finite algebra. Relative to a given set Ω of relations yielding a duality, they characterized the optimal dualities as the dualities determined by the transversals of a certain family of subsets of Ω. However the structure of these subsets — known as globally minimal failsets — remained to be understood. This paper gives a complete description of the globally minimal failsets which do not contain partial endomorphisms, and an algorithmic method to determine them. These results are applied, by way of illustration, to the variety of de Morgan algebras and to two further varieties, one of them an Ockham algebra variety and the other a variety of Heyting algebras. All the globally minimal failsets are determined in each case.

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38

Goodwin, Simon M., Gerhard Röhrle, and Glenn Ubly. "On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type." LMS Journal of Computation and Mathematics 13 (September 2, 2010): 357–69. http://dx.doi.org/10.1112/s1461157009000205.

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AbstractWe consider the finiteW-algebraU(𝔤,e) associated to a nilpotent elemente∈𝔤 in a simple complex Lie algebra 𝔤 of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem forU(𝔤,e), we verify a conjecture of Premet, thatU(𝔤,e) always has a 1-dimensional representation when 𝔤 is of typeG2,F4,E6orE7. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal inU(𝔤) whose associated variety is the coadjoint orbit corresponding to e.

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OWENS, KATE S. "EVERY SHIFT AUTOMORPHISM VARIETY HAS AN INFINITE SUBDIRECTLY IRREDUCIBLE MEMBER." Journal of the Australian Mathematical Society 88, no. 2 (April 2009): 231–38. http://dx.doi.org/10.1017/s1446788709000317.

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AbstractA shift automorphism algebra is one satisfying the conditions of the shift automorphism theorem, and a shift automorphism variety is a variety generated by a shift automorphism algebra. In this paper, we show that every shift automorphism variety contains a countably infinite subdirectly irreducible algebra.

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PAPALEXIOU, NIKOLAOS. "ON THE PRIME SPECTRUM OF THE ENVELOPING ALGEBRA AND CHARACTERISTIC VARIETIES." Journal of Algebra and Its Applications 06, no. 03 (June 2007): 369–83. http://dx.doi.org/10.1142/s0219498807002259.

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Let 𝔤 be a semisimple Lie algebra and U(𝔤), its enveloping algebra. A problem in the theory of non-commutative algebras is the description of the set Spec U(𝔤) of prime ideals in U(𝔤) as topological space. Using the notion of the characteristic variety as introduced by Joseph, we compute some order relations between prime ideals.

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Campercholi, Miguel, Diego Castaño, and José Patricio Díaz Varela. "Algebraic functions in Łukasiewicz implication algebras." International Journal of Algebra and Computation 26, no. 02 (March 2016): 223–47. http://dx.doi.org/10.1142/s0218196716500119.

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In this article we study algebraic functions in [Formula: see text]-subreducts of MV-algebras, also known as Łukasiewicz implication algebras. A function is algebraic on an algebra [Formula: see text] if it is definable by a conjunction of equations on [Formula: see text]. We fully characterize algebraic functions on every Łukasiewicz implication algebra belonging to a finitely generated variety. The main tool to accomplish this is a factorization result describing algebraic functions in a subproduct in terms of the algebraic functions of the factors. We prove a global representation theorem for finite Łukasiewicz implication algebras which extends a similar one already known for Tarski algebras. This result together with the knowledge of algebraic functions allowed us to give a partial description of the lattice of classes axiomatized by sentences of the form [Formula: see text] within the variety generated by the 3-element chain.

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SALEMKAR, ALI REZA, and ZAHRA RIYAHI. "THE POLYNILPOTENT MULTIPLIER OF LIE ALGEBRAS." Journal of Algebra and Its Applications 12, no. 02 (December 16, 2012): 1250154. http://dx.doi.org/10.1142/s021949881250154x.

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Let [Formula: see text] be the variety of polynilpotent Lie algebras of class row (c1, …, cn). The paper is devoted to present the concepts of polynilpotent multiplier [Formula: see text] and cover of a Lie algebra L with respect to the variety [Formula: see text], and compute [Formula: see text] for some known Lie algebras. Also, we determine the structure of all covers of Lie algebras whose polynilpotent multipliers are finite-dimensional, and investigate some common properties between covers.

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IDZIAK, PAWEŁ, KATARZYNA SŁOMCZYŃSKA, and ANDRZEJ WROŃSKI. "FREGEAN VARIETIES." International Journal of Algebra and Computation 19, no. 05 (August 2009): 595–645. http://dx.doi.org/10.1142/s0218196709005251.

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A class [Formula: see text] of algebras with a distinguished constant term 0 is called Fregean if congruences of algebras in [Formula: see text] are uniquely determined by their 0–cosets and ΘA (0,a) = ΘA (0,b) implies a = b for all [Formula: see text]. The structure of Fregean varieties is investigated. In particular it is shown that every congruence permutable Fregean variety consists of algebras that are expansions of equivalential algebras, i.e. algebras that form an algebraization of the purely equivalential fragment of the intuitionistic propositional logic. Moreover the clone of polynomials of any finite algebra A from a congruence permutable Fregean variety is uniquely determined by the congruence lattice of A together with the commutator of congruences. Actually we show that such an algebra A itself can be recovered (up to polynomial equivalence) from its congruence lattice expanded by the commutator, i.e. the structure ( Con (A); ∧, ∨, [·,·]). This leads to Fregean frames, a notion that generalizes Kripke frames for intuitionistic propositional logic.

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44

BELOV-KANEL, A., A. BERZINS, and R. LIPYANSKI. "AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE ASSOCIATIVE ALGEBRA." International Journal of Algebra and Computation 17, no. 05n06 (August 2007): 923–39. http://dx.doi.org/10.1142/s0218196707003901.

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Let [Formula: see text] be the variety of associative algebras over a field K and A = K 〈x1,…, xn〉 be a free associative algebra in the variety [Formula: see text] freely generated by a set X = {x1,…, xn}, End A the semigroup of endomorphisms of A, and Aut End A the group of automorphisms of the semigroup End A. We prove that the group Aut End A is generated by semi-inner and mirror automorphisms of End A. A similar result is obtained for the automorphism group Aut [Formula: see text], where [Formula: see text] is the subcategory of finitely generated free algebras of the variety [Formula: see text]. The later result solves Problem 3.9 formulated in [17].

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DRENSKY, VESSELIN, and JIE-TAI YU. "PRIMITIVE ELEMENTS OF FREE METABELIAN ALGEBRAS OF RANK TWO." International Journal of Algebra and Computation 13, no. 01 (February 2003): 17–33. http://dx.doi.org/10.1142/s021819670300133x.

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Let F(x,y) be a relatively free algebra of rank 2 in some variety of algebras over a field K of characteristic 0. In this paper we consider the problem whether p(x,y) ∈ F(x,y) is a primitive element (i.e. an automorphic image of x): (i) If F(x,y)/(p(x,y)) ≅ F(z), the relatively free algebra of rank 1 (ii) If p(f,g) is primitive for some injective endomorphism (f,g) of F(x,y) (iii) If p(x,y) is primitive in a relatively free algebra of larger rank. These problems have positive solutions for polynomial algebras in two variables. We give the complete answer for the free metabelian associative and Lie algebras and some partial results for free associative algebras.

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JACKSON, MARCEL, and TIM STOKES. "IDENTITIES IN THE ALGEBRA OF PARTIAL MAPS." International Journal of Algebra and Computation 16, no. 06 (December 2006): 1131–59. http://dx.doi.org/10.1142/s0218196706003426.

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We consider the identities of a variety of semigroup-related algebras modelling the algebra of partial maps. We show that the identities are intimately related to a weak semigroup deductive system and we show that the equational theory is decidable. We do this by giving a term rewriting system for the variety. We then show that this variety has many subvarieties whose equational theory interprets the full uniform word problem for semigroups and consequently are undecidable. As a corollary it is shown that the equational theory of Clifford semigroups whose natural order is a semilattice is undecidable.

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Losev, Ivan, and Victor Ostrik. "Classification of finite-dimensional irreducible modules over -algebras." Compositio Mathematica 150, no. 6 (April 7, 2014): 1024–76. http://dx.doi.org/10.1112/s0010437x13007604.

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AbstractFinite $W$-algebras are certain associative algebras arising in Lie theory. Each $W$-algebra is constructed from a pair of a semisimple Lie algebra ${\mathfrak{g}}$ (our base field is algebraically closed and of characteristic 0) and its nilpotent element $e$. In this paper we classify finite-dimensional irreducible modules with integral central character over $W$-algebras. In more detail, in a previous paper the first author proved that the component group $A(e)$ of the centralizer of the nilpotent element under consideration acts on the set of finite-dimensional irreducible modules over the $W$-algebra and the quotient set is naturally identified with the set of primitive ideals in $U({\mathfrak{g}})$ whose associated variety is the closure of the adjoint orbit of $e$. In this paper, for a given primitive ideal with integral central character, we compute the corresponding $A(e)$-orbit. The answer is that the stabilizer of that orbit is basically a subgroup of $A(e)$ introduced by G. Lusztig. In the proof we use a variety of different ingredients: the structure theory of primitive ideals and Harish-Chandra bimodules for semisimple Lie algebras, the representation theory of $W$-algebras, the structure theory of cells and Springer representations, and multi-fusion monoidal categories.

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Hambrook, K., and S. L. Wismath. "Minimal Characteristic Algebras for Rectangular k-Normal Identities." Algebra Colloquium 18, no. 04 (December 2011): 611–28. http://dx.doi.org/10.1142/s1005386711000460.

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A characteristic algebra for a hereditary property of identities of a fixed type τ is an algebra [Formula: see text] such that for any variety V of type τ, we have [Formula: see text] if and only if every identity satisfied by V has the property p. This is equivalent to [Formula: see text] being a generator for the variety determined by all identities of type τ which have property p. Płonka has produced minimal (smallest cardinality) characteristic algebras for a number of hereditary properties, including regularity, normality, uniformity, biregularity, right- and leftmost, outermost, and external-compatibility. In this paper, we use a construction of Płonka to study minimal characteristic algebras for the property of rectangular k-normality. In particular, we construct minimal characteristic algebras of type (2) for k-normality and rectangularity for 1 ≤ k ≤ 3.

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Goguen, Joseph, and Răzvan Diaconescu. "An Oxford survey of order sorted algebra." Mathematical Structures in Computer Science 4, no. 3 (September 1994): 363–92. http://dx.doi.org/10.1017/s0960129500000517.

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This paper surveys several different variants of order sorted algebra (abbreviated OSA), comparing some of the main approaches (overloaded OSA, universe OSA, unified algebra, term declaration algebra, etc.), emphasising motivation and intuitions, and pointing out features that distinguish the original ‘overloaded’ OSA approach from some later developments. These features include sort constraints and retracts; the latter is particularly useful for handling multiple data representations (including automatic coercions among them). Many examples are given, for most of which, runs are shown on the OBJ3 system.This paper also significantly generalises overloaded OSA by dropping the regularity and monotonicity assumptions, and by adding signatures of non-monotonicities, which support simple semantics for some aspects of object oriented programming. A number of new results for this generalisation are proved, including initiality, variety, and quasi-variety theorems. Axiomatisability results à la Birkhoff are also proved for unified algebras.

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BEHN, ANTONIO, ALBERTO ELDUQUE, and ALICIA LABRA. "A CLASS OF LOCALLY NILPOTENT COMMUTATIVE ALGEBRAS." International Journal of Algebra and Computation 21, no. 05 (August 2011): 763–74. http://dx.doi.org/10.1142/s0218196711006455.

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This paper deals with the variety of commutative non associative algebras satisfying the identity [Formula: see text], γ ∈ K. In [3] it is proved that if γ = 0, 1 then any finitely generated algebra is nilpotent. Here we generalize this result by proving that if γ ≠ -1, then any such algebra is locally nilpotent. Our results require characteristic ≠ 2, 3.

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