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As an appropriate generalisation of the features of the classical (Schein) theory of representations of inverse semigroups in $mathscr{I}_{X}$, a theory of representations of inverse semigroups by homomorphisms into complete atomistic inverse algebras is developed. This class of inverse algebras includes partial automorphism monoids of entities such as graphs, vector spaces and modules. A workable theory of decompositions is reached; however complete distributivity is required for results approaching those of the classical case.
We find necessary and sufficient conditions on an (inverse) semigroup $X$ under which its semigroups of maximal linked systems $lambda(X)$, filters $phi(X)$, linked upfamilies $N_2(X)$, and upfamilies $upsilon(X)$ are inverse.
A new construction of a free inverse semigroup was obtained by Poliakova and Schein in 2005. Based on their result, we find a Groebner-Shirshov basis of a free inverse semigroup relative to the deg-lex order of words. In particular, we give the (uniq
This note proves a generalisation to inverse semigroups of Anisimovs theorem that a group has regular word problem if and only if it is finite, answering a question of Stuart Margolis. The notion of word problem used is the two-tape word problem -- t
First we give a definition of a coverage on a inverse semigroup that is weaker than the one gave by a Lawson and Lenz and that generalizes the definition of a coverage on a semilattice given by Johnstone. Given such a coverage, we prove that there ex
A congruence on an inverse semigroup $S$ is determined uniquely by its kernel and trace. Denoting by $rho_k$ and $rho_t$ the least congruence on $S$ having the same kernel and the same trace as $rho$, respectively, and denoting by $omega$ the univers