On the cohomology of the inverse semigroup G of the G-sets of a groupoid G

Renault has defined in [7] the cohomology of the inverse semigroup G of the G-sets of a given groupoid G as a functor from the category of G-presheaves to that of abelian groups. We show in our paper that G-presheaves is isomorphic to AbD(G) where D(G) is the division category defined from Loganathan in [6] and used there to give another description of the Lausch cohomology of inverse semigroups. This isomorphism allows us in turn to prove that Renault and Lausch cohomology groups of G are isomorphic

Fuzzy semigroups via semigroups

The theory of fuzzy semigroups is a branch of mathematics that arose in early 90's as an effort to characterize properties of semigroups by the properties of their fuzzy subsystems which include, fuzzy subsemigroups and their alike, fuzzy one (resp. two) sided ideals, fuzzy quasi-ideals, fuzzy bi-ideals etc. To be more precise, a fuzzy subsemigroup of a given semigroup $(S,\cdot)$ is just a $\wedge$-prehomomorphism $f$ of $(S,\cdot)$ to $([0,1],\wedge)$. Variations of this, which correspond to the other before mentioned fuzzy subsystems, can be obtained by imposing certain properties to $f$. It turns out from the work of Kuroki, Mordeson, Malik and that of many of their descendants, that fuzzy subsystems play a similar role to the structure theory of semigroups that play their non fuzzy analogues. The aim of the present paper is to show that this similarity is not coincidental. As a first step to this, we prove that there is a 1-1 correspondence between fuzzy subsemigroups of $S$ and subsemigroups of a certain type of $S\times I$. Restricted to fuzzy one sided ideals, this correspondence identifies the above fuzzy subsystems to their analogues of $S\times I$. Using these identifications, we prove that the characterization of the regularity of semigroups in terms of fuzzy one sided ideals and fuzzy quasi-ideals can be obtained as an implication of the corresponding non fuzzy analogue

Finiteness Conditions for Clifford Semigroups

There is a large amount of published work in the last decade on finiteness conditionsof monoids and groups such as n FP and its siblings. Recently Gray and Pride havefound that a Clifford monoid containing a minimal idempotent e is of type n FP ifand only if its maximal subgroup containing e is of the same type. In our paper welook for results which are in the same spirit as the above, that is, we try to relate thehomological finiteness conditions of a Clifford monoid to those of a certain grouparising from its semilattice structure. More specifically, we prove that if acommutative Clifford monoid S is of type n FP , then its maximum group image Gis of the same type. To achieve this we employ a result of [10] which relates thecohomology groups of S to those of G, and the fact that the functor n ( , )S Ext   commutes with direct limits whenever S is of type n FP

On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups

In this paper, we develop a technique which enables us to obtain several results from the theory of Γ-semigroups as logical implications of their semigroup theoretical analogues