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groupoid; variety; nonfinitely based
Slim groupoids are groupoids satisfying $x(yz)\=xz$. 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.
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[2] R. McKenzie: Tarski’s finite basis problem is undecidable. Int. J. Algebra and Computation 6 (1996), 49–104. DOI 10.1142/S0218196796000040 | MR 1371734 | Zbl 0844.08011
[3] R. McKenzie, G. McNulty and W. Taylor: Algebras, Lattices, Varieties, Volume I. Wadsworth & Brooks/Cole, Monterey, CA, 1987. MR 0883644
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