Congruences on glrac semigroups (I)

The theory of congruences on semigroups is an important part in the theory of semigroups. The aim of this paper is to study [Formula: see text]-congruences on a glrac semigroup. It is proved that the [Formula: see text]-congruences on a glrac semigroup become a complete sublattice of its lattice of congruences. Especially, the structures of left restriction semigroup [Formula: see text]-congruences and the projection-separating [Formula: see text]-congruences on a glrac semigroup are established. Also, we demonstrate that they are both complete sublattice of [Formula: see text]-congruences and consider their relations with respect to complete lattice homomorphism.

Dually hemimorphic semi-Nelson algebras

Extending the relation between semi-Heyting algebras and semi-Nelson algebras to dually hemimorphic semi-Heyting algebras, we introduce and study the variety of dually hemimorphic semi-Nelson algebras and some of its subvarieties. In particular, we prove that the category of dually hemimorphic semi-Heyting algebras is equivalent to the category of dually hemimorphic centered semi-Nelson algebras. We also study the lattice of congruences of a dually hemimorphic semi-Nelson algebra through some of its deductive systems.

On Lattice-Ordered Rees Matrix Γ-Semigroups

The purpose of this paper is to introduce and give some properties of l-Rees matrix Γ-semigroups. Generalizing the results given by Guowei and Ping, concerning the congruences and lattice of congruences on regular Rees matrix Γ-semigroups, the structure theorem of l-congruences lattice of l - Γ-semigroup M = μº(G : I; L; Γe) is given, from which it follows that this l-congruences lattice is distributive.

The Lattice of Congruences on a Subdirectly Irreducible Weak Stone-Ockham Algebra

Weak Stone-Ockham algebras are those algebras (L; ∧, ∨, f, ⋆,0,1) of type 〈2,2,1,1,0,0〉, where (L; ∧, ∨, f,0,1) is an Ockham algebra, (L; ∧, ∨, ⋆,0,1) is a weak Stone algebra, and the unary operations f and ⋆ commute. In this paper, we give a complete description of the structure of the lattice of congruences on the subdirectly irreducible algebras.

Quasi-subtractive varieties

Varieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras. Abstract algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member A of a τ-regular variety the lattice of congruences of A is isomorphic to the lattice of deductive filters on A of the τ-assertional logic of . Moreover, if has a constant 1 in its type and is 1-subtractive, the deductive filters on A ∈ of the 1-assertional logic of coincide with the -ideals of A in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation.However, there are isomorphism theorems, for example, in the theories of residuated lattices, pseudointerior algebras and quasi-MV algebras that cannot be subsumed by these general results. The aim of the present paper is to appropriately generalise the concepts of subtractivity and τ-regularity in such a way as to shed some light on the deep reason behind such theorems. The tools and concepts we develop hereby provide a common umbrella for the algebraic investigation of several families of logics, including substructural logics, modal logics, quantum logics, and logics of constructive mathematics.

VARIETIES OF DIFFERENTIAL MODES EMBEDDABLE INTO SEMIMODULES

Differential modes provide examples of modes that do not embed as subreducts into semimodules over commutative semirings. The current paper studies differential modes, so-called Szendrei differential modes, which actually do embed into semimodules. These algebras form a variety. The main result states that the lattice of nontrivial subvarieties is dually isomorphic to the (nonmodular) lattice of congruences of the free commutative monoid on two generators. Consequently, all varieties of Szendrei differential modes are finitely based.

Balanced Double Ockham Algebras

For a wide class of double Ockham algebras we characterise the subdirectly irreducible members by determining the structure of their lattice of congruences.