Periods of Algebraic Varieties

A variety of algebras is a Specht variety if each of its subvarieties is finitely based. The problem of whether the variety of soluble alternative algebras is a Specht variety was formulated by A. M. Slin′ko in “Dnestrovskaya Tetrad′” [1, Problem 129]. This problem was solved in the affirmative in the case of a field of characteristic other than 2 and 3. As regards index 2 soluble algebras, this follows from the results of Yu. A. Medvedev’s article [2]. Furthermore, in [3] Yu. A. Medvedev exhibited a variety of soluble alternative algebras over a field of characteristic 2 without a finite basis of identities. Next, let Nk denote the variety of alternative algebras of nilpotency class at most k and A, the variety of algebras with zero multiplication. S. V. Pchelintsev [4] proved that every soluble alternative algebra A over a field of characteristic other than 2 and 3 belongs to the variety NkA ∩N3Nm; i.e., (A) = (A) = 0

Contents Introduction § 1. Varieties of linear algebras § 2. Residually nilpotent chain varieties of algebras § 3. Precomplete varieties of algebras § 4. Chain varieties of alternative, right alternative Lie-admissible, and Jordan algebras § 5. Chain varieties of restricted Lie p-algebras § 6. Varieties of associative algebras with a distributive lattice of subvarieties § 7. Lattices of certain varieties of linear algebras § 8. Lattices of varieties of algebraic systems Conclusion References

We introduce a variety of algebras in the language of Boolean algebras with an extra implication, namely the variety of pseudo-subordination algebras, which is closely related to subordination algebras. We believe it provides a minimal general algebraic framework where to place and systematise the research on classes of algebras related to several kinds of subordination algebras. We also consider the subvariety of pseudo-contact algebras, related to contact algebras, and the subvariety of the strict implication algebras introduced in Bezhanishvili et al. [(2019). A strict implication calculus for compact Hausdorff spaces. Annals of Pure and Applied Logic, 170, 102714]. The variety of pseudo-subordination algebras is term equivalent to the variety of Boolean algebras with a binary modal operator. We exploit this fact in our study. In particular, to obtain a topological duality from which we derive the known topological dualities for subordination algebras and contact algebras.

We show that varieties of algebras over abstract clones and over the corresponding operads are rationally equivalent. We introduce the class of operads (which we call commutative for definiteness) such that the varieties of algebras over these operads resemble in a sense categories of modules over commutative rings. In particular, the notions of a polylinear mapping and the tensor product of algebras. The categories of modules over commutative rings and the category of convexors are examples of varieties over commutative operads. By analogy with the theory of linear multioperator algebras, we develop a theory of C-linear multioperator algebras; in particular, of algebras, defined by C-polylinear identities (here C is a commutative operad). We introduce and study symmetric C-linear operads. The main result of this article is as follows: A variety of C-linear multioperator algebras is defined by C-polylinear identities if and only if it is rationally equivalent to a variety of algebras over a symmetric C-linear operad.

Varieties of algebras

Wedderburn theorem on varieties of algebras

Let K be a field, char K ≠ 2, and let X be a finite set, X = {x1, ... , x n }. In what follows, F = F(X) denotes the free K-algebra without the unity element on the set X of free generators of one of the following varieties of algebras over a field K: the variety of all algebras, the variety of Lie algebras, varieties of color Lie superalgebras, the variety of Lie p-algebras, varieties of color Lie p-superalgebras, and varieties of commutative and anticommutative algebras.

A monoidSisright coherentif every finitely generated subact of every finitely presented rightS-act is finitely presented. This is the non-additive notion corresponding to that for a ringRstating that every finitely generated submodule of every finitely presented rightR-module is finitely presented. For monoids (and rings) right coherency is an important finitary property which determines, amongst other things, the existence of amodel companionof the class of rightS-acts (rightR-modules) and hence that the class of existentially closed rightS-acts (rightR-modules) is axiomatisable.Choo, Lam and Luft have shown that free rings are right (and left) coherent; the authors, together with Ruškuc, have shown that (free) groups, free commutative monoids and free monoids have the same properties. It is then natural to ask whether other free algebras in varieties of monoids, possibly with an augmented signature, are right coherent. We demonstrate that free inverse monoids are not.Munn described the free inverse monoid FIM(Ω) on Ω as consisting of birooted finite connected subgraphs of the Cayley graph of the free group on Ω. Sitting within FIM(Ω) we have free algebras in other varieties and quasi-varieties, in particular the free left ample monoid FLA(Ω) and the free ample monoid FAM(Ω). The former is the free algebra in the variety of unary monoids corresponding to partial maps with distinguished domain; the latter is the two-sided dual. For example, FLA(Ω) is obtained from FIM(Ω) by considering only subgraphs with vertices labelled by elements of the free monoid on Ω.The main objective of the paper is to show that FLA(Ω)isright coherent. Furthermore, by making use of the same techniques we show that FIM(Ω), FLA(Ω) and FAM(Ω) satisfy (R), (r), (L) and (l), related conditions arising from the axiomatisability of certain classes of rightS-acts and of leftS-acts.

A variety of algebras is called distinguished if there is a countably generated, locally finite algebra such that any other countably generated locally finite algebra is a homomorphic image of . This article continues the investigation of the question of when a variety of associative algebras is distinguished.For example, if the ground field is uncountable, then every distinguished variety is nonmatric. Note that nonmatric varieties over an algebraically closed field are always distinguished and, over a field of characteristic zero, a nonmatric variety is distinguished if and only if , where is the algebraic closure of .Bibliography: 16 titles.

We consider a couple of versions of the classical Kurosh problem (whether there is an infinite-dimensional algebraic algebra) for varieties of linear multioperator algebras over a field. We show that, given an arbitrary signature, there is a variety of algebras of this signature such that the free algebra of the variety contains polylinear elements of arbitrarily large degree, while the clone of every such element satisfies some nontrivial identity. If, in addition, the number of binary operations is at least 2, then each such clone may be assumed to be finite-dimensional. Our approach is the following: we cast the problem in the language of operads and then apply the usual homological constructions in order to adopt Golod’s solution to the original Kurosh problem. This paper is expository, so that some proofs are omitted. At the same time, the general relations of operads, algebras, and varieties are widely discussed.

In this article, we review results on primitive elements of free algebras of main types of Schreier varieties of algebras. A variety of linear algebras over a field is Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras. A system of elements of a free algebra is primitive if it is a subset of some set of free generators of this algebra. We consider free nonassociative algebras, free commutative and anti-commutative nonassociative algebras, free Lie algebras and superalgebras, and free Lie p-algebras and p-superalgebras. We present matrix criteria for systems of elements of elements. Primitive elements distinguish automorphisms: endomorphisms sending primitive elements to primitive elements are automorphisms. We give a series of examples of almost primitive elements (an element of a free algebra is almost primitive if it is not a primitive element of the whole algebra, but it is a primitive element of any proper subalgebra which contains it). We also consider generic elements and Δ-primitive elements.

We say that an algebra \({\mathcal{A}}\) has the retraction closure property (RCP) if the set of all retractions of \({\mathcal{A}}\) is closed with respect to fundamental operations of \({\mathcal{A}}\) applied pointwise. In this paper we investigate this property, both “locally” (one algebra) and “globally” (in some variety of algebras), especially emphasizing the case of groupoids. We compare the retraction closure property with the endomorphism closure property on both levels and prove that a necessary and sufficient condition for a variety V of algebras to have RCP is that V is a variety of entropic algebras that satisfy the diagonal identities.

The calculus of classes and closure operations has proved to be a useful tool in group theory and has led to a deep theory in the study of finite soluble groups. More recently, parallel theories have started to be developed in various varieties of algebras, such as Lie, Leibniz and Malcev algebras. This paper seeks to investigate the extent to which these later theories can be generalized to the variety of all non-associative algebras.

Representability of monoids by endomorphisms of algebras from a given class C is often decided by a category-theoretical approach: a full embedding of a binding category B into C is constructed. Since every category of algebras is isomorphic to a full subcategory of any binding category and each small category occurs as a full subcategory of B, every monoid is then isomorphic to the endomorphism monoid of an algebra in C. Furthermore, by the Hedrlfn-KuSera Theorem [19], under the set-theoretical axiom (M), saying that there is only a set of strongly measurable cardinals, any concrete category can be fully embedded into any binding category B. There are numerous binding varieties of algebras. Thus, for instance, Hedrl in and Pultr [7] proved that a category of algebras of a given type zl is binding ff and only if the sum of Zl is bigger than 1. The list includes semigroups [8], commutative rings with unit [3, 4], lattices with (0, 1)-homomorphisms [6], and a locally finite lattice variety [2], to name a few. This leads to a problem of characterization of binding varieties. Two non-structural characterizations are given in Rosicl@ [23] where it is shown that under (M) every concrete complete, cocomplete, locally and colocally small category is binding if and only if a certain two-object category can be fully embedded into it, and in Sichler [26], where the representability of a finite category similarly decides whether a variety of unary algebras is binding. A structural theorem appears in Pultr and Sichler [21] where all varieties of idempotent unary algebras with two operations which are binding are described by identities. The aim of this paper is a characterization of binding (and almost-binding) varieties among those definable by identities in basic operations only. We prove: let V be a variety of algebras of type A given by identities containing only basic operations. Then V is almost-binding if and only if the sum of arities of operations in A is bigger than 1 and it is binding if and only if it is

Associative algebras over an associative commutative ring with unity are considered. A variety of algebras is said to be permutative if it satisfies an identity of the form x 1 x 2⋯x n = x 1σ x 2σ ⋯x nσ , where σ is a nontrivial permutation on a set {1, 2, … , n}. Minimal elements in the lattice of all nonpermutative varieties are called almost permutative varieties. By Zorn’s lemma, every nonpermutative variety contains an almost permutative variety as a subvariety. We describe almost permutative varieties of algebras over a finite field and almost commutative varieties of rings. In [5], such varieties were characterized for the case of algebras over an infinite field.