×

Slim groupoids. (English) Zbl 1161.20055

Summary: Slim groupoids are groupoids satisfying \(x(yz)\approx xz\). We find all simple slim groupoids and all minimal varieties of slim groupoids. Every slim groupoid can be embedded into a subdirectly irreducible slim groupoid. The variety of slim groupoids has the finite embeddability property, so that the word problem is solvable. We introduce the notion of a strongly nonfinitely based slim groupoid (such groupoids are inherently nonfinitely based) and find all strongly nonfinitely based slim groupoids with at most four elements; up to isomorphism, there are just two such groupoids.

MSC:

20N02 Sets with a single binary operation (groupoids)
08B15 Lattices of varieties
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] T. Evans: Embeddability and the word problem. J. London Math. Soc. 28 (1953), 76–80. · Zbl 0050.02801 · doi:10.1112/jlms/s1-28.1.76
[2] R. McKenzie: Tarski’s finite basis problem is undecidable. Int. J. Algebra and Computation 6 (1996), 49–104. · Zbl 0844.08011 · doi:10.1142/S0218196796000040
[3] R. McKenzie, G. McNulty and W. Taylor: Algebras, Lattices, Varieties, Volume I. Wadsworth & Brooks/Cole, Monterey, CA, 1987. · Zbl 0611.08001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.