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Register a new userIntra-regular Abel-Grassmanns groupoids
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We characterize intra-regular Abel-Grassmanns groupoids by the properties of their ideals and $(in ,in!vee q_{k})$-fuzzy ideals of various types.
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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.
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.
Antilattices $(S;lor, land)$ for which the Greens equivalences $mathcal L_{(lor)}$, $mathcal R_{(lor)}$, $mathcal L_{(land)}$ and $mathcal R_{(land)}$ are all congruences of the entire antilattice are studied and enumerated.
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.
Let A be an associative algebra over an algebraically closed field F of characteristic zero and let G be a finite abelian group. Regev and Seeman introduced the notion of a regular G-grading on A, namely a grading A= {Sigma}_{g in G} A_g that satisfies the following two conditions: (1) for every integer n>=1 and every n-tuple (g_1,g_2,...,g_n) in G^n, there are elements, a_i in A_{g_i}, i=1,...,n, such that a_1*a_2*...*a_n != 0. (2) for every g,h in G and for every a_g in A_g,b_h in A_h, we have a_{g}b_{h}=theta(g,h)b_{h}a_{g}. Then later, Bahturin and Regev conjectured that if the grading on A is regular and minimal, then the order of the group G is an invariant of the algebra. In this article we prove the conjecture by showing that ord(G) coincides with an invariant of A which appears in PI theory, namely exp(A) (the exponent of A). Moreover, we extend the whole theory to (finite) nonabelian groups and show that the above result holds also in that case.
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