Compactness of systems of equations in semigroups



: Harju T, Karhumaki J, Plandowski W

: 1995

Lecture Notes in Computer Science

AUTOMATA, LANGUAGES AND PROGRAMMING

: LECT NOTES COMPUT SC

: 944

: 444

: 454

: 11

: 3-540-60084-1

: 0302-9743


We consider systems u(i) = v(i) (i epsilon I) of equations in semigroups over finite sets of variables. A semigroup S is said to satisfy the compactness property (or CP, for short), if each system of equations has an equivalent finite subsystem. It is shown that all monoids in a variety V satisfy CP, if and only if the finitely generated monoids in V satisfy the maximal condition on congruences. We also show that if a finitely generated semigroup S satisfies CP, then S is necessarily hopfian and satisfies the chain condition on idempotents. Finally, we give three simple examples (the bicyclic monoid, the free monogenic inverse semigroup and the Baumslag-Solitar group) which do not satisfy CP, and show that the above necessary conditions are not sufficient.



Last updated on 2025-13-10 at 12:27