Abstract
Let us call measure of complexity over a class ℋ of semigroups any function β from IN into IN such that any semigroup H in ℋ generated by at most n elements has its cardinality less than β(n). If β is a measure of complexity over the class of all subgroups of a finite matrix semigroup H, then we can compute an integer T(β) which is greater than the cardinality of H. We give here equations computing effectively such an integer T(β) for some classes of matrix semigroups, or of quotients semigroups of matrix semigroups.
As a consequence we prove, using a theorem of A.I. Kostrikin, that there exist an effective decision procedure for the finiteness of matrix semigroups over skewfields, under the condition that each of its elements admits a given prime integer p as period.
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Jacob, G. (1977). Complexite des demi — Groupes de matrices. In: Salomaa, A., Steinby, M. (eds) Automata, Languages and Programming. ICALP 1977. Lecture Notes in Computer Science, vol 52. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-08342-1_21
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DOI: https://doi.org/10.1007/3-540-08342-1_21
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