وثيقة
Commutative semigroups whose endomorphisms are power functions.
المعرف
DOI: 10.1007/s00233-021-10178-x
المصدر
Semigroup Forum. v. 102, 3, p. 737-755
الدولة
Germany.
مكان النشر
Berlin
الناشر
Springer.
ميلادي
2021-06-01
اللغة
الأنجليزية
الملخص الإنجليزي
For any commutative semigroup S and positive integer m the power function f: S→ S defined by f(x) = xm is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.
ISSN
0037-1912
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