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Register a new userCompletely inverse $AG^{**}$-groupoids
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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.
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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.
We characterize intra-regular Abel-Grassmanns groupoids by the properties of their ideals and $(in ,in!vee q_{k})$-fuzzy ideals of various types.
It is proved that if A_p is a countable elementary abelian p-group, then: (i) The ring End(A_p) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End(A_p)/I, where I is the ideal of End(A_p) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End(A_p) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of the endomorphism rings of modules over commutative rings is also obtained.
We give a concise introduction to (discrete) algebras arising from etale groupoids, (aka Steinberg algebras) and describe their close relationship with groupoid C*-algebras. Their connection to partial group rings via inverse semigroups also explored.
Several complete congruences on the lattice L(CR) of varieties of completely regular semi- groups have been fundamental to studies of the structure of L(CR). These are the kernel relation K , the left trace relation Tl , the right trace relation Tr and their intersections KcapTl,Kcap Tr . However, with the exception of the lattice of all band varieties which happens to coincide with the kernel class of the trivial variety, almost nothing is known about the internal structure of individual K-classes beyond the fact that they are intervals in L(CR). Here we present a number of general results that are pertinent to the study of K -classes. This includes a variation of the renowned Polak Theorem and its relationship to the complete retraction V -> V cap B , where B denotes the variety of bands. These results are then applied, here and in a sequel, to the detailed analysis of certain families of K -classes. The paper concludes with results hinting at the complexity of K -classes in general, such as that the classes of relation K/Kl may have the cardinality of the continuum.
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