- 1.
- Embedding semigroups in groups: A geometrical approach, Publ. Inst. Math.
Beograd 38 (1985), 69-82.
- 2.
- On a theorem of Shutov, Publ. Inst. Math. Beograd 38 (1985), 83-85.
- 3.
- Dependence among quasi-identities axiomatizing the class of semigroups
embeddable in groups, abstract (1987), in preparation.

*SUMMARY:*

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
.
If we take the semigroup presented (on
generators
)
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
*x*_{i}*y*_{j}=*x*_{k}*y*_{l} 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 *S*^{2}.) Mal'cev's original treatment
is purely algebraic and the definition of his axioms is
quite complicated. He considers group words over the
alphabet
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 *S*implies 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.

*REFERENCES:*

- [B]
- L. A. Bokut', Embeddings of rings, Russian
Math. Surveys, 42:4 (1987), 105-138.
- [L]
- J. Lambek, The immersibility of a semigroup into a
group, Canad. J. Math. 3 (1951), 34-43.
- [M]
- A. I. Mal'cev,
On the immersion of associative systems into groups I,
*II*, Mat. Sbornik 6 (1939), 331-336; 8 (1940), 241-164. - [P]
- M. Peixoto, On the classification of flows on
2-manifolds, in: Dynamical Systems, Academic Press: 1973.