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Register a new userCongruences on completely inverse $AG^{**}$-groupoids
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By a completely inverse $AG^{**}$-groupoid we mean an inverse $AG^{**}$-groupoid $A$ satisfying the identity $xx^{-1}=x^{-1}x$, where $x^{-1}$ denotes a unique element of $A$ such that $x=(xx^{-1})x$ and $x^{-1}=(x^{-1}x)x^{-1}.$ We show that the set of all idempotents of such groupoid forms a semilattice and the Greens relations $mathcal{H,L, R,D}$ and $mathcal{J}$ coincide on $A$. The main result of this note says that any completely inverse $AG^{**}$-groupoid meets the famous Lallements Lemma for regular semigroups. Finally, we show that the Greens relation $mathcal{H}$ is both the least semilattice congruence and the maximum idempotent-separating congruence on any completely inverse $AG^{**}$-groupoid.
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A completely inverse $AG^{**}$-groupoid is a groupoid satisfying the identities $(xy)z=(zy)x$, $x(yz)=y(xz)$ and $xx^{-1}=x^{-1}x$, where $x^{-1}$ is a unique inverse of $x$, that is, $x=(xx^{-1})x$ and $x^{-1}=(x^{-1}x)x^{-1}$. First we study some fundamental properties of such groupoids. Then we determine certain fundamental congruences on a completely inverse $AG^{**}$-groupoid; namely: the maximum idempotent-separating congruence, the least $AG$-group congruence and the least $E$-unitary congruence. Finally, we investigate the complete lattice of congruences of a completely inverse $AG^{**}$-groupoids. In particular, we describe congruences on completely inverse $AG^{**}$-groupoids by their kernel and trace.
This paper enriches the list of properties of the congruence sequences starting from the universal relation and successively performing the operations of lower $t$ and lower $k$. Three classes of completely regular semigroups, namely semigroups for which $ker{sigma}$ is a cryptogroup, semigroups for which $ker{ u}$ is a cryptogroup and semigroups for which $kappa$ is over rectangular bands, are studied. $((omega_t)_k)_t$, $((mathcal{D}_t)_k)_t$ and $((omega_k)_t)_k$ are found to be the least congruences on $S$ such that the quotient semigroups are semigroups for which $ker{sigma}$ is a cryptogroup, $ker{ u}$ is a cryptogroup and $kappa$ is over rectangular bands, respectively. The results obtained present a response to three problems in Petrich and Reillys textbook textquotelefttextquoteleft Completely Regular Semigroupstextquoterighttextquoteright.
We discuss some types of congruences on Menger algebras of rank $n$, which are generalizations of the principal left and right congruences on semigroups. We also study congruences admitting various types of cancellations and describe their relationship with strong subsets.
We characterize intra-regular Abel-Grassmanns groupoids by the properties of their ideals and $(in ,in!vee q_{k})$-fuzzy ideals of various types.
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 exists a pseudogroup that is universal in the sense that it transforms cover-to-join idempotent-pure maps into idempotent-pure pseudogroup homomorphisms. Then, we show how to go from a nucleus on a pseudogroup to a topological groupoid embedding of the corresponding groupoids. Finally, we apply the results found to study Exels notions of tight filters and tight groupoids.
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