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On some varieties of symmetric idempotent entropic groupoids. (English) Zbl 0762.20024
Universal and applied algebra, Proc. 5th Symp., Turawa/Pol. 1988, 254-274 (1989).
[For the entire collection see Zbl 0733.00008.]
A symmetric idempotent entropic groupoid is a groupoid satisfying the identities \((xy)y=x\), \(xx=x\) and \((xy)(zt)=(xz)(yt)\). The lattice of varieties of SIE groupoids has been described by the author in an earlier paper. The aim of the present paper is to describe the varieties of SIE groupoids that are equivalent to some variety of quasigroups (it turns out that these are equivalent to varieties of abelian groups); free groupoids in such varieties are described, and there is a result on the structure of groupoids in varieties of SIE groupoids containing a variety equivalent to a variety of abelian groups. An application in graph theory is given.
Reviewer: J.Ježek (Praha)

MSC:
20N02 Sets with a single binary operation (groupoids)
08B15 Lattices of varieties
08B20 Free algebras
20N05 Loops, quasigroups
20M07 Varieties and pseudovarieties of semigroups
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