20,630 research outputs found

    Completely inverse AGβˆ—βˆ—AG^{**}-groupoids

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    A completely inverse AGβˆ—βˆ—AG^{**}-groupoid is a groupoid satisfying the identities (xy)z=(zy)x(xy)z=(zy)x, x(yz)=y(xz)x(yz)=y(xz) and xxβˆ’1=xβˆ’1xxx^{-1}=x^{-1}x, where xβˆ’1x^{-1} is a unique inverse of xx, that is, x=(xxβˆ’1)xx=(xx^{-1})x and xβˆ’1=(xβˆ’1x)xβˆ’1x^{-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 AGβˆ—βˆ—AG^{**}-groupoid; namely: the maximum idempotent-separating congruence, the least AGAG-group congruence and the least EE-unitary congruence. Finally, we investigate the complete lattice of congruences of a completely inverse AGβˆ—βˆ—AG^{**}-groupoids. In particular, we describe congruences on completely inverse AGβˆ—βˆ—AG^{**}-groupoids by their kernel and trace

    Limits over categories of extensions

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    We consider limits over categories of extensions and show how certain well-known functors on the category of groups turn out as such limits. We also discuss higher (or derived) limits over categories of extensions.Comment: 18 page
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