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Each combinatorial
strict inverse semigroup S is determined by (1) a partially ordered set X
which in fact is the partially ordered set of the ℐ-classes of S, (2) pairwise
disjoint sets Iα indexed by the elements of X which in fact form the collection
of 𝒟- (equivalently: ℐ-) related idempotents and (3) structure mappings
fα,β : Iα → Iβ for α ≥ β satisfying certain compatibility conditions. The
multiplication on S can be described in terms of the parameters X, Iα, fα,β.
Conversely, the system (X;Iα,fα,β) can be characterized abstractly in order that it
defines a uniquely determined combinatorial strict inverse semigroup. In
this paper, the constituting parameters X, Iα, fα,β of the combinatorial
strict inverse free product S of a collection of combinatorial strict inverse
semigroups Si are described in terms of the parameters of the semigroups
Si.
As an application it is shown that the word problem for such a free product in
general is not decidable.
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