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arXiv · 1305.6856

Completely inverse $AG^{**}$-groupoids

Wieslaw A. Dudek·Roman S. Gigoń
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Abstract

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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Wieslaw A. Dudek, Roman S. Gigoń. 2013-05-25. Completely inverse $AG^{**}$-groupoids. https://doi.org/10.1007/s00233-013-9465-z

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