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Vietnam Journal of Mathematics 35:4(2007) 495-505  Finite Semigroups with Infinite Product and Languages of Infinite Words Phan Trung Huy, Nguyen Quy Khang,and Do Long Van Abstract.  A semigroup (monoid) $S$, which can be equipped with an infinite product compatible with the given multiplication in $S$, is called a semigroup (monoid, resp.) with infinite product (abbreviated by SWIP and MWIP, resp.). In this paper, necessary and sufficient conditions for a finite semigroup to be a SWIP are established. Relationships between MWIPs and varieties of finite monoids, and also between MWIPs and regular languages of infinite words are considered.   2000 Mathematics Subject Classification: 20M35, 20M07, 68Q45, 68Q68. Keywords: Semigroup, monoid, infinite product, variety of finite monoids, $\omega$-language.

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