On residuated skew lattices

In this paper, we define residuated skew lattice as non-commutative generalization of residuated lattice and investigate its properties. We show that Green’s relation 𝔻 is a congruence relation on residuated skew lattice and its quotient algebra is a residuated lattice. Deductive system and skew deductive system in residuated skew lattices are defined and relationships between them are given and proved. We define branchwise residuated skew lattice and show that a conormal distributive residuated skew lattice is equivalent with a branchwise residuated skew lattice under a condition.

On the coset structure of a skew lattice

The class of skew lattices can be seen as an algebraic category. It models an algebraic theory in the category of sets where the Green’s relation

Ideals and Green's relations in ordered semigroups

Exactly as in semigroups, Green's relations play an important role in the theory of ordered semigroups—especially for decompositions of such semigroups. In this paper we deal with theℐ-trivial ordered semigroups which are defined via the Green's relationℐ, and with the nil andΔ-ordered semigroups. We prove that every nil ordered semigroup isℐ-trivial which means that there is no ordered semigroup which is 0-simple and nil at the same time. We show that in nil ordered semigroups which are chains with respect to the divisibility ordering, every complete congruence is a Rees congruence, and that this type of ordered semigroups are△-ordered semigroups, that is, ordered semigroups for which the complete congruences form a chain. Moreover, the homomorphic images of△-ordered semigroups are△-ordered semigroups as well. Finally, we prove that the ideals of a nil ordered semigroupSform a chain under inclusion if and only ifSis a chain with respect to the divisibility ordering.

Green's Relation ℛ on the Monoid of Clone Endomorphisms

A hypersubstitution is a map which takes n-ary operation symbols to n-ary terms. Any such map can be uniquely extended to a map defined on the set Wτ(X) of all terms of type τ, and any two such extensions can be composed in a natural way. Thus, the set Hyp (τ) of all hypersubstitutions of type τ forms a monoid. In this paper, we characterize Green's relation ℛ on the monoid Hyp (τ) for the type τ=(n,n). In this case, the monoid of all hypersubstitutions is isomorphic with the monoid of all clone endomorphisms. The results can be applied to mutually derived varieties.

Completely Regular Semigroups with Generalized Strong Semilattice Decompositions

The concept of ρG-strong semilattice of semigroups is introduced. By using this concept, we study Green's relation ℋ on a completely regular semigroup S. Necessary and sufficient conditions for S/ℋ to be a regular band or a right quasi-normal band are obtained. Important results of Petrich and Reilly on regular cryptic semigroups are generalized and enriched. In particular, characterization theorems of regular cryptogroups and normal cryptogroups are obtained.

THE ORBIT STRUCTURE OF FINITE MONOIDS

Let M be a finite monoid with unit group G. We consider a refinement, [Formula: see text] of the Green’s relation [Formula: see text]. The [Formula: see text]-classes, denoted [Formula: see text] are the G×G orbits, GHG, of the ℋ-classes, H, of M. With an orbit [Formula: see text] we associate a local monoid [Formula: see text] and determine the structure of these local monoids. The theory is applied to the full transformation semigroup [Formula: see text] and we see that the number of orbits [Formula: see text] in [Formula: see text] is equal to the number of partitions of n.