Narrow Results

In this paper, we study the geometric equivalence of algebras in several varieties of algebras. We solve some of the problems formulated in [2], in particular, that of geometric equivalence for real-closed fields and finitely generated commutative groups.

We consider universal algebras which are monoids and which have a binary operation we call internalized equality, satisfying some natural conditions. We show that the class of such E-structures has a characterization in terms of a distinguished submonoid which is a semilattice. Some important varieties (and variety-like classes) of E-structures are considered, including E-semilattices (which we represent in terms of topological spaces), E-rings (which we show are equivalent to rings with a generalized interior operation), E-quantales (where internalized equalities on a fixed quantale in which 1 is the largest element are shown to correspond to sublocales of the quantale), and EI-structures (in which an internalized inequality is defined and interacts in a natural way with the equality operation).

A semiring with involution is a semiring equipped with an involutorial antiasutomorphism as a fundamental operation. The aim of the present paper is to determine the lattice of all varieties of idempotent and distributive semirings with involution. We start with the description of their structure, which is followed by a complete list of all subdirectly irreducibles. We make a heavy use of general results obtained recently by Dolinka and Vinčić [11] on involutorial Płonka sums. Applying these results and some further structural theorems, we construct the considered lattice. It turns out that it has exactly 64 elements.

It is proved that the lattice L(B_d) of quasivarieties contained in the variety B_d of idempotent semigroups contains an isomorphic copy of the ideal lattice of a free lattice on ω free generators. This result shows that a problem of Petrich [19], which calls for a description of L(B_d), is much more complex than originally expected.

Using techniques pioneered by R. McKenzie, we prove that there is no algorithm which, given a finite algebra in a finite language, determines whether the variety (equational class) generated by the algebra has a model companion. In particular, there exists a finite algebra such that the variety it generates has no model companion; this answers a question of Burris and Werner from 1979.

For a partially ordered set P, let Co(P) denote the lattice of all order-convex subsets of P. For a positive integer n, we denote by (resp., SUB(n)) the class of all lattices that can be embedded into a lattice of the form

(1) Both classes and SUB(n), for any positive integer n, are locally finite, finitely based varieties of lattices, and we find finite equational bases of these varieties.

(2) The variety is the quasivariety join of all the varieties SUB(n), for 1≤n<ω, and it has only countably many subvarieties. We classify these varieties, together with all the finite subdirectly irreducible members of .

(3) Every finite subdirectly irreducible member of is projective within , and every subquasivariety of is a variety.

Let W(X) be a free commutative or a free associative algebra. The group of automorphisms of the semigroup End(W(X)) is studied.

Let V be a non-trivial variety of bounded distributive lattices with a quantifier, as introduced by Cignoli in [7]. It is shown that if V does not contain the 4-element bounded Boolean lattice with a simple quantifier, then V contains non-isomorphic algebras with isomorphic endomorphism monoids, but there are always at most two such algebras. Further, it is shown that if V contains the 4-element bounded Boolean lattice with a simple quantifier, then it is finite-to-finite universal (in the categorical sense) and, as a consequence, for any monoid M, there exists a proper class of non-isomorphic algebras in V for which the endomorphism monoid of every member is isomorphic to M.

We find a syntactic characterization of the class of lattices embeddable into convexity lattices of posets which are trees. The characterization implies that this class forms a finitely based variety.

We say that a finite algebra 𝔸 = 〈A; F〉 has the ability to count if there are subalgebras C of 𝔸³ and Z of 𝔸 such that the structure 〈A; C, Z〉 has the ability to count in the sense of Feder and Vardi. We show that for a core relational structure A the following conditions are equivalent: (i) the variety generated by the algebra 𝔸 associated to A contains an algebra with the ability to count; (ii) 𝔸² has the ability to count; (iii) the variety generated by 𝔸 admits the unary or affine type. As a consequence, for CSP's of finite signature, the bounded width conjectures stated in Feder–Vardi [10], Larose–Zádori [17] and Bulatov [5] are identical.

We consider the problem, whether the algebras in two finitely generated congruence-distributive varieties have isomorphic congruence lattices. According to the results of P. Gillibert, this problem is closely connected with the question, which diagrams of finite distributive semilattices can be represented by the congruence lattices of algebras in a given variety. We study this question for varieties of bounded lattices, generated by different nondistributive lattices of length 2 (denoted M_n). For each pair from this family of varieties we construct a diagram indexed by the product of three finite chains, which is liftable in one variety and nonliftable in the other one. We also discover an interesting link to the four-color theorem of graph theory.

A limit variety is a variety that is minimal with respect to being nonfinitely based. This paper presents a new infinite series of limit semigroup varieties, each of which is generated by a finite 0-simple semigroup with Abelian subgroups. These varieties exhaust all limit varieties generated by completely 0-simple semigroups with Abelian subgroups.

By adjusting a method of Kadourek and Polák developed for free semigroups satisfying x^r ≏ x, we prove that if is a periodic group variety, then any maximal subgroup of the free object in the completely regular semigroup variety of the form is a relatively free group in over a suitable set of free generators. When is locally finite, we provide some bounds for the sizes of its finitely generated members.

Constructions that yield pseudorecursiveness in [I] (Int. J. Algebra Comput.6 (1996) 457–510) are extended in this article. Finitely based varieties of semigroups with increasingly strict expansions by additional unary operation symbols or individual constants are shown to have the pseudorecursive property: the equational theory is undecidable, but the subsets obtained by bounding the number of distinct variables are all recursive. The most stringent case considered here is the single unary operation or distinguished element. New techniques of stratified reducibility and interpretation via rewriting rules are employed to show the property inherits along a chain of theories. Pure semigroup varieties that are both finitely based and pseudorecursive will be discussed in a later paper.

We study the class of finite lattices that are isomorphic to the congruence lattices of algebras from a given finitely generated congruence-distributive variety. If this class is as large as allowed by an obvious necessary condition, the variety is called congruence FD-maximal. The main results of this paper characterize some special congruence FD-maximal varieties.

We prove that the class K(σ) of all algebraic structures of signature σ is Q-universal if and only if there is a class K ⊆ K(σ) such that the problem whether a finite lattice embeds into the lattice of K-quasivarieties is undecidable.

We study connections between closure operators on an algebra (A, Ω) and congruences on the extended power algebra defined on the same algebra. We use these connections to give an alternative description of the lattice of all subvarieties of semilattice ordered algebras.

We study varieties of associative algebras over a finite field and varieties of associative rings satisfying semigroup or adjoint semigroup identities. We characterize these varieties in terms of “forbidden algebras” and discuss some corollaries of the characterizations.

An $n$-ary operation $f$ is called symmetric if, for all permutations $\pi$ of $\{1,\dots ,n\}$, it satisfies the identity $f(x_1,x_2,\dots ,x_n)=f(x_{\pi (1)},x_{\pi (2)},\dots ,x_{\pi (n)})$. We show that, for each finite algebra ${𝒜}$, either it has symmetric term operations of all arities or else some finite algebra in the variety generated by ${𝒜}$ has two automorphisms without a common fixed point. We also show this two-automorphism condition cannot be replaced by a single fixed-point-free automorphism.

We study the structure and properties of free skew Boolean algebras (SBAs). For finite generating sets, these free algebras are finite and we give their representation as a product of primitive algebras and provide formulas for calculating their cardinality. We also characterize atomic elements and central elements, and calculate the number of such elements. These results are used to study minimal generating sets of finite SBAs. We also prove that the center of the free infinitely generated algebra is trivial and show that all free algebras have intersections.