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Axiomatization for semigroups embeddable in groups

Embedding semigroups in groups: A geometrical approach, Publ. Inst. Math. Beograd 38 (1985), 69-82.

On a theorem of Shutov, Publ. Inst. Math. Beograd 38 (1985), 83-85.

Dependence among quasi-identities axiomatizing the class of semigroups embeddable in groups, abstract (1987), in preparation.


Whether every cancellative semigroup can be embedded in a group was a question posed in the thirties by van der Waerden and soon answered in the negative by A. I. Mal'cev [M]. Mal'cev's observation was that every group (and hence every embeddable semigroup) satisfies the quasi-identity $x_1y_1=x_2=y_2\land
x_3y_1=x_4y_2\land x_3y_3=x_4y_4\Rightarrow
x_1y_3=x_2y_4$. If we take the semigroup presented (on generators $x_1,\ldots ,y_4$) by the relators occuring in the antecedent part of this quasi-identity, then it will be cancellative, but the equality in the consequent part of the quasi-identity will not hold in it. Besides exhibiting this example, Mal'cev actually proved much more: The class of embeddable semigroups is not finitely axiomatizable. He also gave an infinite axiomatization and another was later found by J. Lambek [L]. In [1], I give a geometric interpretation of Mal'cev and Lambek axioms. Both Malcev's and Lambek's axioms are quasi-identities of the form ``a conjuction of equalities of the form xiyj=xkyl implies an identity of the same form''. The axioms are also quadratic, in the sense that every variable occurs exactly twice. When we draw the van Kampen diagram of such a quasi-identity, we obtain an oriented graph on the sphere, each vertex being either a source, a sink, or a ``monkey saddle'' (edges successively come in and go out from it). Mal'cev's axioms are characterized by the property that every vertex of the third type has the valence 4, and Lambek's axioms are characterized as those for which all sinks have valence 2. The same characterization was independently proved by V. N. Gerasimov [B]. (In other words, van Kampen diagrams of Mal'cev's axioms are precisely the phase portraits of gradient-like flows on S2.) Mal'cev's original treatment is purely algebraic and the definition of his axioms is quite complicated. He considers group words over the alphabet ${x_1,\ldots ,x_p,y_1,\ldots,y_q}$ such that every letter and the inverse of every letter occur exactly once; moreover, deleting all the letters of the same type (x or y) should result in a trivial group word (totally cancellable). We can call such words ``shuffles of quadratic words of genus zero''; to every such shuffle Mal'cev associates an axiom. He also observes that if a shuffle S' is obtained from a shuffle S by deleting some letters, then the quasi-identity associated to Simplies the one associated to S'. Mal'cev asked whether this relation fully describes the logical dependence among his axioms and in [3] I prove that this is essentially true. I haven't written this down yet. As an interesting by-product, I obtain a description and classification of gradient-like flows on surfaces by ``shuffles of quadratic words''. This classification has some advantages over the known ones, given by Peixoto [P] and others.


L. A. Bokut', Embeddings of rings, Russian Math. Surveys, 42:4 (1987), 105-138.

J. Lambek, The immersibility of a semigroup into a group, Canad. J. Math. 3 (1951), 34-43.

A. I. Mal'cev, On the immersion of associative systems into groups I, II, Mat. Sbornik 6 (1939), 331-336; 8 (1940), 241-164.

M. Peixoto, On the classification of flows on 2-manifolds, in: Dynamical Systems, Academic Press: 1973.

next up previous
Next: Quadratic quasigroup varieties Up: SUMMARY OF RESULTS Previous: Enumeration of quadratic words