Narrow Results

We describe the inherently non-dualisable finite algebras from some semigroup related classes. The classes for which this problem is solved include the variety of bands, the pseudovariety of aperiodic monoids, commutative monoids, and (assuming a reasonable conjecture in the literature) the varieties of all finite monoids and finite inverse semigroups. The first example of an inherently non-dualisable entropic algebra is also presented.

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.

A translation in an algebraic signature is a finite conjunction of equations in one variable. On a quasivariety K, a translation τ naturally induces a deductive system, called the τ-assertional logic of K. Two quasivarieties are τ-assertionally equivalent if they have the same τ-assertional logic. This paper is a study of assertional equivalence. It characterizes the quasivarieties equivalent to ones with various desirable properties, such as τ-regularity (a general form of point regularity). Special attention is paid to structural properties of quasivarieties that are assertionally equivalent to their varietal closures under an indicated translation.

A quasivariety is a universal Horn class of algebraic structures containing the trivial structure. The set of all subquasivarieties of a quasivariety forms a complete lattice under inclusion. A lattice isomorphic to for some quasivariety is called a lattice of quasivarieties or a quasivariety lattice. The Birkhoff–Maltsev Problem asks which lattices are isomorphic to lattices of quasivarieties. A lattice L is called unreasonable if the set of all finite sublattices of L is not computable, that is, there is no algorithm for deciding whether a finite lattice is a sublattice of L. The main result of this paper states that for any signature σ containing at least one non-constant operation, there is a quasivariety of signature σ such that the quasivariety lattice is unreasonable. Moreover, there are uncountable unreasonable lattices of quasivarieties. We also present some corollaries of the main result.

We show that for every quasivariety 𝒦 of structures (where both functions and relations are allowed) there is a semilattice S with operators such that the lattice of quasi-equational theories of 𝒦 (the dual of the lattice of sub-quasivarieties of 𝒦) is isomorphic to Con(S, +, 0, 𝒡. As a consequence, new restrictions on the natural quasi-interior operator on lattices of quasi-equational theories are found.

Part I proved that for every quasivariety 𝒦 of structures (which may have both operations and relations) there is a semilattice S with operators such that the lattice of quasi-equational theories of 𝒦 (the dual of the lattice of sub-quasivarieties of 𝒦) is isomorphic to Con(S, +, 0, 𝒡). It is known that if S is a join semilattice with 0 (and no operators), then there is a quasivariety 𝒬 such that the lattice of theories of 𝒬 is isomorphic to Con(S, +, 0). We prove that if S is a semilattice having both 0 and 1 with a group 𝒢 of operators acting on S, and each operator in 𝒢 fixes both 0 and 1, then there is a quasivariety 𝒲 such that the lattice of theories of 𝒲 is isomorphic to Con(S, +, 0, 𝒢).

Convex subsets of affine spaces over the field of real numbers are described by so-called barycentric algebras. In this paper, we discuss extensions of the geometric and algebraic definitions of a convex set to the case of more general coefficient rings. In particular, we show that the principal ideal subdomains of the reals provide a good framework for such a generalization. Since the closed intervals of these subdomains play an essential role, we provide a detailed analysis of certain cases, and discuss differences from the "classical" intervals of the reals. We introduce a new concept of an algebraic closure of "geometric" convex subsets of affine spaces over the subdomains in question, and investigate their properties. We show that this closure provides a purely algebraic description of topological closures of geometric generalized convex sets. Our closure corresponds to one instance of the very general closure introduced in an earlier paper of the authors. The approach used in this paper allows to extend some results from that paper. Moreover, it provides a very simple description of the closure, with concise proofs of existence and uniqueness.

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.

A finite unary algebra of finite type with a constant function 0 that is a one-element subalgebra, and whose operations have range {0, 1}, is called a {0, 1}-valued unary algebra with 0. Such an algebra has a finite basis for its quasi-equations if and only if the relation defined by the rows of the nontrivial functions in the clone form an order ideal.

We determine all quasivarieties of aperiodic semigroups that are contained in some residually finite variety. This endeavor was initially motivated by a problem in natural dualities, but our work here also serves as a partial correction to an error found in a result of Sapir from the 1980s.

A subdirect product $A\le \mathrm{\Pi }〈A_i:i\in I〉$ is global if there is a topology on $I$ which makes of $A$ the algebra of all global sections of a sheaf whose stalks are the algebras $A_i$ with $i\in I$. A quasivariety ${𝒬}$ has BL-global representations if there is a class ${ℱ}$ containing but close to the class of all relatively subdirectly irreducible members of ${𝒬}$, satisfying that every member of ${𝒬}$ is isomorphic to a global subdirect product whose factors are in ${ℱ}$. The adjective “BL” refers to “Birkhoff-like” since this type of representations are analogous to the classical Birkhoff subdirect representation by relatively subdirectly irreducibles. This paper has two main contributions. The first one is a generalization of a theorem proved in [D. Vaggione, Sheaf representation and Chinese remainder theorems, Algebra Universalis 29 (1992) 232–272] which characterizes the existence of a global representation of an algebra in terms of the solvability of certain congruence systems. The second one is a theorem assuring the existence of BL-global representations for quasivarieties with a near unanimity term.

We characterize the universal theory of the free Burnside groups of large prime exponent and briefly show how to extend and modify our results to arbitrary sufficiently large odd exponent.