Narrow Results

The Congruence Lattice Problem (CLP), stated by R. P. Dilworth in the forties, asks whether every distributive {∨, 0}-semilattice S is isomorphic to the semilattice Con_c L of compact congruences of a lattice L.

While this problem is still open, many partial solutions have been obtained, positive and negative as well. The solution to CLP is known to be positive for all S such that |S|≤ℵ₁. Furthermore, one can then take Lwith permutable congruences. This contrasts with the case where |S|≥ℵ₂, where there are counterexamples S for which Lcannot be, for example, sectionally complemented. We prove in this paper that the lattices of these counterexamples cannot have permutable congruences as well.

We also isolate finite, combinatorial analogues of these results. All the "finite" statements that we obtain are amalgamation properties of the Con_c functor. The strongest known positive results, which originate in earlier work by the first author, imply that many diagrams of semilattices indexed by the square2² can be lifted with respect to the Con_c functor.

We prove that the latter results cannot be extended to the cube, 2³. In particular, we give an example of a cube diagram of finite Boolean semilattices and semilattice embeddings that cannot be lifted, with respect to the Con_c functor, by lattices with permutable congruences.

We also extend many of our results to lattices with almost permutable congruences, that is, α∨β=αβ ∪βα, for all congruences α and β.

We conclude the paper with a very short proof that no functor from finite Boolean semilattices to lattices can lift the Con_c functor on finite Boolean semilattices.

An involuted semilattice <S,∨,^-> is a semilattice <S,∨> with an involution ^-: S→S, i.e., <S,∨,^-> satisfies , and . In this paper we study the properties of such semilattices. In particular, we characterize free involuted semilattices in terms of ordered pairs of subsets of a set. An involuted semilattice <S,∨,^-,1> with greatest element 1 is said to be complemented if it satisfies a∨ā=1. We also characterize free complemented semilattices. We next show that complemented semilattices are related to ternary algebras. A ternary algebra <T,+,*,^-,0,ϕ,1> is a de Morgan algebra with a third constant ϕ satisfying , and (a+ā)+ϕ=a+ā. If we define a third binary operation ∨ on T as a∨b=a*b+(a+b)*ϕ, then <T,∨,^-,ϕ> is a complemented semilattice.

We construct a diagram , indexed by a finite partially ordered set, of finite Boolean 〈∨, 0, 1〉-semilattices and 〈∨, 0, 1〉-embeddings, with top semilattice 2⁴, such that for any variety V of algebras, if has a lifting, with respect to the congruence lattice functor, by algebras and homomorphisms in V, then there exists an algebra U in V such that the congruence lattice of U contains, as a 0,1-sublattice, the five-element modular nondistributive lattice M₃. In particular, V has an algebra whose congruence lattice is neither join- nor meet-semidistributive Using earlier work of K. A. Kearnes and Á. Szendrei, we also deduce that V has no nontrivial congruence lattice identity.

In particular, there is no functor Φ from finite Boolean semilattices and 〈∨, 0, 1〉-embeddings to lattices and lattice embeddings such that the composition Con Φ is equivalent to the identity (where Con denotes the congruence lattice functor), thus solving negatively a problem raised by P. Pudlák in 1985 about the existence of a functorial solution of the Congruence Lattice Problem.

We prove that for every distributive 〈∨,0〉-semilattice S, there are a meet-semilattice P with zero and a map μ: P × P → S such that μ(x,z) ≤ μ(x,y) ∨ μ(y,z) and x ≤ y implies that μ(x,y) = 0, for all x, y, z ∈ P, together with the following conditions:.

(P1) μ(v,u) = 0 implies that u = v, for all u ≤ v in P.

(P2) For all u ≤ v in P and all a, b ∈ S, if μ(v,u) ≤ a ∨ b, then there are a positive integer n and a decomposition u = x₀ ≤ x₁ ≤ ⋯ ≤ x_n = v such that either μ(x_{i + 1}, x_i) ≤ a or μ(x_i+1, x_i) ≤ b, for each i < n.

(P3) The subset {μ(x,0) | x ∈ P} generates the semilattice S.

Furthermore, every finite, bounded subset of P has a join, and P is bounded in case S is bounded. Furthermore, the construction is functorial on lattice-indexed diagrams of finite distributive 〈∨,0,1〉-semilattices.

We study what kinds of limits are preserved by the greatest semilattice image functor from the category of all semigroups to its subcategory of all semilattices.

For a class of algebras, denote by Con_c the class of all (∨, 0)-semilattices isomorphic to the semilattice Con_cA of all compact congruences of A, for some A in . For classes and of algebras, we denote by the smallest cardinality of a (∨, 0)-semilattices in Con_c which is not in Con_c if it exists, ∞ otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruence-distributive varieties and , is either finite, or ℵ_n for some natural number n, or ∞. We also find two finitely generated modular lattice varieties and such that , thus answering a question by J. Tůma and F. Wehrung.

Hedrlín and Pultr proved that for any monoid M there exists a graph G with endomorphism monoid isomorphic to M. In a previous paper, we give a construction G(M) for a graph with prescribed endomorphism monoid M known as a -graph. Using this construction, we derived bounds on the minimum number of vertices and edges required to produce a graph with a given endomorphism monoid for various classes of finite monoids. In this paper, we generalize the -graph construction and derive several new bounds for monoid classes not handled by our first paper. Among these are the so called "strong semilattices of C-semigroups" where C is one of the following: Groups, Abelian Groups, Rectangular Groups, and completely simple semigroups.

We present a new solution of the word problem of free algebras in varieties generated by iterated semidirect products of semilattices. As a consequence, we provide asymptotical bounds for free spectra of these varieties. In particular, each finite -trivial (and, dually, each finite -trivial) semigroup has a free spectrum whose logarithm is bounded above by a polynomial function.

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).

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.

A finite lattice may be regarded as a join semilattice with $0$. Using this viewpoint, we give algorithms for testing semidistributivity, provide a new characterization of convex geometries, and characterize congruence lattices of finite join semidistributive lattices.

Much of the structure theory of inverse semigroups is based on constructing arbitrary inverse semigroups from groups and semilattices. It is known that E-unitary (or proper) inverse semigroups may be described as P-semigroups (McAlister), or inverse subsemigroups of semidirect products of a semilattice by a group (O'Carroll) or C_u-semigroups built over an inverse category acted upon by a group (Margolis and Pin). On the other hand, every inverse semigroup is known to have an E-unitary inverse cover (McAlister).

The aim of this paper is to develop a similar theory for proper weakly left ample semigroups, a class with properties echoing those of inverse semigroups. We show how the structure of semigroups in this class is based on constructing semigroups from unipotent monoids and semilattices. The results corresponding to those of McAlister, O'Carroll and Margolis and Pin are obtained.

The main concern of this paper is with the equations satisfied by the algebra of truth values of type-2 fuzzy sets. That algebra has elements all mappings from the unit interval into itself with operations given by certain convolutions of operations on the unit interval. There are a number of positive results. Among them is a decision procedure, similar to the method of truth tables, to determine when an equation holds in this algebra. One particular equation that holds in this algebra implies that every subalgebra of it that is a lattice is a distributive lattice. It is also shown that this algebra is locally finite. Many questions are left unanswered. For example, we do not know whether or not this algebra has a finite equational basis, that is, whether or not there is a finite set of equations from which all equations satisfied by this algebra follow. This and various other topics about the equations satisfied by this algebra will be discussed.

We study the relationship between numerical semigroups and periodic subadditive functions.

Let M be a Clifford monoid and let θ be an endomorphism of M. We prove that if the Bruck–Reilly extension BR(M, θ) is finitely presented then M is finitely generated. This allows us to derive necessary and sufficient conditions for Bruck–Reilly extensions of Clifford monoids to be finitely presented.

The set of all endomorphisms of an algebraic structure with composition of functions as operation is a rich source of semigroups, which has only rarely been dipped into (see [2]). Here we make a start by considering endomorphisms of Clifford semigroups, relating them to the homomorphisms and endomorphisms of the underlying groups.

Let be the ring of Laurent polynomials in commuting variables. As a generalization of the toroidal Lie algebra, the gradation shifting toroidal Lie algebra is isomorphic to the corresponding (centerless) toroidal Lie algebra so(n, ℂ) ⨂ A of type B or D as a vector space, with the Lie bracket twisted by n fixed elements E₁,…,E_n from A. In this paper, we study the automorphisms of the gradation shifting toroidal algebra , which is proved to be closely related to a class of subgroups of GL(n,ℤ), called the linear groups over semilattices. We use the linear group over a special semilattice to determine the automorphism group of the gradation shifting toroidal algebra , which extends our earlier work.

We modify the concept of SL-homomorphism introduced by K.P. Shum, P. Zhu and N. Kehayopulu [7] for directed posets. Since every directed poset can be converted into an algebra called a directoid, we investigate the relationship between our homomorphisms and these of directoids. We point out that no modification of SL-congruences is necessary since congruences induced by our homomorphisms are just that of directoids.

A semigroup S is called a ∇-semigroup if the set of all its full subsemigroups forms a chain under set inclusion. In this paper, we investigate some properties of such kind of semigroups and establish several characterization theorems of type-A ∇-semigroups. Our theorems generalize the known results of P. R. Jones obtained in 1981 on inverse semigroups whose full inverse subsemigroups form a chain.