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Register a new userRelationships among quasivarieties induced by the min networks on inverse semigroups
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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 universal congruence on $S$, we consider the sequence $omega$, $omega_k$, $omega_t$, $(omega_k)_t$, $(omega_t)_k$, $((omega_k)_t)_k$, $((omega_t)_k)_t$, $cdots$. The quotients ${S/omega_k}$, ${S/omega_t}$, ${S/(omega_k)_t}$, ${S/(omega_t)_k}$, ${S/((omega_k)_t)_k}$, ${S/((omega_t)_k)_t}$, $cdots$, as $S$ runs over all inverse semigroups, form quasivarieties. This article explores the relationships among these quasivarieties.
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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 (unique and shortest) Groebner-Shirshov normal forms in the classes of equivalent words of a free inverse semigroup together with the Groebner-Shirshov algorithm to transform any word to its normal form.
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 -- the set of all pairs of words over a generating set for the semigroup which both represent the same element.
Let $X$ be a nonempty set and $X^{2}$ be the Cartesian square of $X$. Some semigroups of binary relations generated partitions of $X^2$ are studied. In particular, the algebraic structure of semigroups generated by the finest partition of $X^{2}$ and, respectively, by the finest symmetric partition of $X^{2}$ are described.
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