Weak congruences of an algebra with the CEP and the WCIP

Here we consider the weak congruence lattice of an algebra with the congruence extension property (the CEP for short) and the weak congruence intersection property (briefly the WCIP). In the first section we give necessary and sufficient conditions for the semimodularity of that lattice. In the second part we characterize algebras whose weak congruences form complemented lattices.     

半群的模糊同余扩张

本文引入半群的模糊同余扩张的概念,给出了模糊同余扩张的同态性质,同时,本文研究了带有模糊同余扩张性质的半群类,证明了一个半群S有模糊同余扩张性质当且仅当S有同余扩张性质,最后进一步给出 有模糊同余张张性质的半群类的特征。     

Semigroups with the Congruence Extension Property

ρT on a semigroup of T of S extends to the semigroup S, if there exists a congruence _ρ on s such that _{ρ|T= ρT}. A semigroup S has the congruence extension property, CEP, if each congruence on each semigroup extends to S. In this paper we characterize the semigroups with CEP by a set of conditions on their structure (by this we answer a problem put forward in [1]). In particular, every such semigroup is a semilattice of nil extensions of rectangular groups.     

Congruence Permutable Symmetric Extended de Morgan Algebras

An algebra A is said to be congruence permutable if any two congruences on it are permutable. This property has been investigated in several varieties of algebras, for example, de Morgan algebras, p-algebras, Kn,0-algebras. In this paper, we study the class of symmetric extended de Morgan algebras that are congruence permutable. In particular we consider the case where A is finite, and show that A is congruence permutable if and only if it is isomorphic to a direct product of finitely many simple algebras.     

Uniform Mal’cev algebras with small congruence lattices

A congruence of an algebra is called uniform if all the congruence classes are of the same size. An algebra is called uniform if each of its congruences is uniform. All algebras with a group reduct have this property. We prove that almost every finite uniform Mal’cev algebra with a congruence lattice of height at most two is polynomially equivalent to an expanded group.     

Ω-compact semigroups having congruence extension property

The congruence extension property (CEP) of semigroups has been extensively studied by a number of authors. We call a compact semigroup S an Ω-compact semigroup if the set of all regular elements of S forms an ideal of S. In this note, we characterize the Ω-compact semigroup having (CEP). Our result extends a recent result obtained by X.J. Guo on the congruence extension property of strong Ω-compact semigroups which is a semigroup containing precisely one regular D-class.     

模糊半群上的模糊同余及其扩张

给出模糊半群上的模糊同余的概念，并进一步研究它的一些基本代数性质。同时研究带有模糊半群上的模糊同余扩张性质(FCEPF)的半群类，得到一个半群有模糊半群上的模糊同余扩张性质、有模糊同余扩张性质(FCEP)、有同余扩张性质(CEP)三个条件是等价的。     

On schemes for congruence distributivity

We present diagrammatic schemes characterizing congruence 3-permutable and distributive algebras. We show that a congruence 3-permutable algebra is congruence meetsemidistributive if and only if it is distributive. We characterize varieties of algebras satisfying the so-called triangular scheme by means of a Maltsev-type condition.     

Modular varieties with the Fraser-Horn property

The notion of central idempotent elements in a ring can be easily generalized to the setting of any variety with the property that proper subalgebras are always nontrivial. We will prove that if such a variety is also congruence modular, then it has factorable congruences, i.e., it has the Fraser-Horn property. (This property is well known to have major implications for the structure theory of the algebras in the variety.)

     

The Endomorphism Kernel Property in Finite Distributive Lattices and de Morgan Algebras

Abstract

An algebra 𝒜 has the endomorphism kernel property if every congruence on 𝒜 different from the universal congruence is the kernel of an endomorphism on 𝒜. We first consider this property when 𝒜 is a finite distributive lattice, and show that it holds if and only if 𝒜 is a cartesian product of chains. We then consider the case where 𝒜 is an Ockham algebra, and describe in particular the structure of the finite de Morgan algebras that have this property.     

On the congruence extension property for compact semigroups

A topological semigroupS is said to have thecongruence extension property (CEP) provided that for each closed subsemigroupT ofS and each closed congruence σ onT, σ can be extended to a closed congruence onS. (That is, ∩(T xT=σ). The main result of this paper gives a characteriation of Γ-compact commutative archimedean semigroups with the congruence extension property (CEP). Consideration of this result was motivated by the problem of characterizing compact commutative semigroups with CEP as follows. It is well known that every commutative semigroup can be expressed as a semilattice of archimedean components each of which contains at most one idempotemt. The components of a compact commutative semigroup need not be compact (nor Γ-compact) as the congruence providing the decomposition is not necessarily closed. However, any component with CEP which is Γ-compact is characterized by the afore-mentioned result. Characterization of components of a compact commutative semigroup having CEP is a natural step towar characterization of the entire semigroup since CEP is a hereditary property. Other results prevented in this paper give a characterization of compact monothetic semigroups with CEP and show that Rees quotients of compact semigroups with CEP retain CEP.     

A characterization of congruence permutable locally finite varieties

A variety of universal algebras is said to be congruence permutable if for every algebra A of and every pair of congruences α, β from A we have αβ = βα. We show that if is locally finite (i.e., every finitely generated member of is finite) then congruence permutability is equivalent to a local property of the finite members of , expressible in the language of tame congruence theory. This answers a question of R. McKenzie and D. Hobby.     

Quasiorder lattices of varieties

The set \({{\mathrm{Quo}}}(\mathbf {A})\) of compatible quasiorders (reflexive and transitive relations) of an algebra \(\mathbf {A}\) forms a lattice under inclusion, and the lattice \({{\mathrm{Con}}}(\mathbf {A})\) of congruences of \(\mathbf {A}\) is a sublattice of \({{\mathrm{Quo}}}(\mathbf {A})\). We study how the shape of congruence lattices of algebras in a variety determine the shape of quasiorder lattices in the variety. In particular, we prove that a locally finite variety is congruence distributive [modular] if and only if it is quasiorder distributive [modular]. We show that the same property does not hold for meet semi-distributivity. From tame congruence theory we know that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of congruence lattices isomorphic to the lattice \(\mathbf {M}_3\). We prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for quasiorder lattices of infinite algebras even in the variety of semilattices.     

Projective algebras and primitive subquasivarieties in varieties with factor congruences

We prove that in the varieties where every compact congruence is a factor congruence and every nontrivial algebra contains a minimal subalgebra, a finitely presented algebra is projective if and only if it has every minimal algebra as its homomorphic image. Using this criterion of projectivity, we describe the primitive subquasivarieties of discriminator varieties that have a finite minimal algebra embedded in every nontrivial algebra from this variety. In particular, we describe the primitive quasivarieties of discriminator varieties of monadic Heyting algebras, Heyting algebras with regular involution, Heyting algebras with a dual pseudocomplement, and double-Heyting algebras.     

Congruence properties of pseudocomplemented De Morgan algebras

In this paper we first describe the Priestley duality for pseudocomplemented De Morgan algebras by combining the known dualities of distributive p‐algebras due to Priestley and for De Morgan algebras due to Cornish and Fowler. We then use it to characterize congruence‐permutability, principal join property, and the property of having only principal congruences for pseudocomplemented De Morgan algebras. The congruence‐uniform pseudocomplemented De Morgan algebras are also described.     

Some applications of higher commutators in Mal’cev algebras

We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher commutators to prove that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence modular variety, affine completeness is a decidable property. Moreover, we show that in such algebras, we can check in polynomial time whether two given polynomial terms induce the same function.     

Congruence extension,Hamiltonian and Abelian properties in locally finite varieties

We prove that in a locally finite variety with the congruence extension property, locally solvable congruences are central and locally solvable algebras are Hamiltonian. Also, we prove that a maximal subuniverse of a finite algebra in an Abelian variety is identical with an equivalence class of some congruence.Presented by H. P. Gumm.     

Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class

By a congruence distributive quasivariety we mean any quasivarietyK of algebras having the property that the lattices of those congruences of members ofK which determine quotient algebras belonging toK are distributive. This paper is an attempt to study congruence distributive quasivarieties with the additional property that their classes of relatively finitely subdirectly irreducible members are axiomatized by sets of universal sentences. We deal with the problem of characterizing such quasivarieties and the problem of their finite axiomatizability.Presented by Joel Berman.To the memory of Basia Czelakowska.