An Embedding of an Ordered Groupoid into an le-Groupoid

In this paper we prove that each ordered groupoid (resp. semigroup) S embeds in a complete distributive le-groupoid (resp. le-semigroup) using the ordered groupoid (resp. semigroup) arising from S by the adjunction of a zero element.     

A Combinatorial Property for Semigroups of Matrices

Abstract. In this paper we study the semigroups of matrices over a commutative semiring. We prove that a semigroup of matrices over a tropical semiring satisfies a combinatorial property called weak permutation property . We consider an application of this result to the Burnside problem for groups.     

The Relation B in le-Semigroups

We study the relation B in le-semigroups that mimics the relation "to generate the same bi-ideal" considered by Kapp [1] in the plain semigroup case. This relation in general is finer than the Green-Kehayopulu relation H we studied in [6] and turns out to have better properties. In particular, the regularity (intra-regularity) of an element induces the regularity (respectively intra-regularity) of the whole B-class containing that element. We also provide various conditions under which B-classes are subsemigroups.     

A combinatorial property for semigroups of matrices

In this paper we study the semigroups of matrices over a commutative semiring. We prove that a semigroup of matrices over a tropical semiring satisfies a combinatorial property called weak permutation property. We consider an application of this result to the Burnside problem for groups.     

The Green-Kehayopulu relation H in le-semigroups

The definition of the Green—Kehayopulu relation H in le -semigroups mimics the definition of the usual Green relation H in plain semigroups. We show, however, that certain properties of H -classes essentially differ from those of H-classes: a non-singleton H -class cannot be a subgroup and an H -class H satisfying ``Green's condition' (there exist a,b ∈ H such that a, b ∈ H) need not constitute a subsemigroup. We provide various conditions that ensure that an H -class forms a subsemigroup.     

Clifford semigroups as functors and their cohomology

We prove that the category of Clifford semigroups and prehomomorphisms CSP\mathcal{CSP} is isomorphic to a certain subcategory of the category of diagrams over groups. Under this isomorphism, Clifford semigroups are identified with certain functors. As an application of the isomorphism theorem, we show that the category with objects commutative inverse semigroups having the same semilattice of idempotents and with morphisms, the inverse semigroup homomorphisms that fix the semilattice, imbeds into a category of right modules over a certain ring. Also we find a very close relationship between the cohomology groups of a commutative inverse monoid and the cohomology groups of the colimit group of the functor giving the monoid.     

On some homotopical and homological properties of monoid presentations

If a monoid S is given by some finite complete presentation ℘, we construct inductively a chain of CW-complexes