| 词条 | Aperiodic semigroup |
| 释义 |
In mathematics, an aperiodic semigroup is a semigroup S such that every element x ∈ S is aperiodic, that is, for each x there exists a positive integer n such that xn = xn + 1.[1] An aperiodic monoid is an aperiodic semigroup which is a monoid. Finite aperiodic semigroupsA finite semigroup is aperiodic if and only if it contains no nontrivial subgroups, so a synonym used (only?) in such contexts is group-free semigroup. In terms of Green's relations, a finite semigroup is aperiodic if and only if its H-relation is trivial. These two characterizations extend to group-bound semigroups.{{cn|date=October 2012}} A celebrated result of algebraic automata theory due to Marcel-Paul Schützenberger asserts that a language is star-free if and only if its syntactic monoid is finite and aperiodic.[2] A consequence of the Krohn–Rhodes theorem is that every finite aperiodic monoid divides a wreath product of copies of the three-element flip-flop monoid, consisting of an identity element and two right zeros. The two-sided Krohn–Rhodes theorem alternatively characterizes finite aperiodic monoids as divisors of iterated block products of copies of the two-element semilattice. See also
References1. ^{{cite book | title=Monoids, Acts and Categories: With Applications to Wreath Products and Graphs. A Handbook for Students and Researchers | volume=29 | series=De Gruyter Expositions in Mathematics | first1=Mati | last1=Kilp | first2=Ulrich | last2=Knauer | first3=Alexander V. | last3=Mikhalev | publisher=Walter de Gruyter | year=2000 | isbn=3110812908 | zbl=0945.20036 | page=29 }} 2. ^Schützenberger, Marcel-Paul, "On finite monoids having only trivial subgroups," Information and Control, Vol 8 No. 2, pp. 190–194, 1965.
1 : Semigroup theory |
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