This article is about equations over semigroups. For semigroups compact with respect to a topology, see Topological semigroup. For semigroups of compact operators, see C0-semigroup § Compact semigroups.
In mathematics, a compact semigroup is a semigroup in which the sets of solutions to equations can be described by finite sets of equations. The term "compact" here does not refer to any topology on the semigroup.
Let S be a semigroup and X a finite set of letters. A system of equations is a subset E of the Cartesian product X∗ × X∗ of the free monoid (finite strings) over X with itself. The system E is satisfiable in S if there is a map f from X to S, which extends to a semigroup morphism f from X+ to S, such that for all (u,v) in E we have f(u) = f(v) in S. Such an f is a solution, or satisfying assignment, for the system E.[1]
Two systems of equations are equivalent if they have the same set of satisfying assignments. A system of equations if independent if it is not equivalent to a proper subset of itself.[1] A semigroup is compact if every independent system of equations is finite.[2]
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