Nasir Sohaol
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Dominions and zigzags for monoids
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Let U be a submonoid of a monoid S. Then by the dominion of
U in S we mean the set of all elements d of S such that for every pair of
(monoid) homomorphisms f,g from S to T that agree on U one has f(d) = g(d).
In 1965 J.R. Isbell proved in his famous (zigzag) theorem that d belongs to
the dominion of U in S if and only if there exists a specific system of
equalities which he termed zigzag. Another version of the theorem was put
forward by B. Stenström in 1971, in which zigzags were replaced by certain
tensor equalities. One of the most recent proofs of this theorem is due to
J. Renshaw that adheres to its formulation due to Stenström. My objective
is to acquaint the audience with the techniques used by Renshaw to
establish the said proof. This will provide a base for my subsequent talk
concerning an ordered analogue of the zigzag theorem.

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References

1. Isbell J.R.: Epimorphisms and dominions, proceedings of the
conference on categorical algebra, La Jolla, California, 1965, Springer,
New York.

2. Stenström B.: Flatness and localization over monoids, Math. Nachr.
48-Band, 315-334 (1971).

3. Renshaw J.: On free products of semigroups and a new proof of
Isbell's zigzag theorem, J. Algebra 251, 12-15 (2002).