Mutually divisible semigroups

✍ Scribed by Ronald V. Book; Michael A. Harrison

• Publisher: Elsevier Science
• Year: 1974
• Tongue: English
• Weight: 354 KB
• Volume: 9
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(74)90079-x

✦ Synopsis

In the study of abstract automata, prtrticularly algebraic decomrositisn theory, tho scrfion af "division" of semigroups plays an important rofe Me [ S] )_ A semigroup I? is .said to &vi& a semigroup G if H is a homomorphic image of a subsemigroup of c. )iert;t we arc csncr;mod with semigroups which are mutuaEly divisible, that is, U divides G and G divides H.

If two finite semigruups arc mutually divisible, then they are isomorphic. However, if two Snfinitc scmigroups are mutually divisible, then they IICC~ nat be isomorphic. This hasi been shown by Kaplansky 161 in the case of groups. The purpose of this note is to investigate this phensrtwnon far the case of semigroups. Our first fact is stated forrnall~ in Theorem 1.