Equations on the semidirect product of a finite semilattice by a finite commutative monoid By: Francine Blanchet-Sadri and Xin-Hong Zhang Communicated by: F. J. Pastijn F. Blanchet-Sadri and X.-H. Zhang, "Equations on the Semidirect Product of a Finite Semilattice by a Finite Commutative Monoid." Semigroup Forum, Vol. 49, No. 1, 1994, pp 67-81. Made available courtesy of Springer-Verlag: The original publication is available at http://www.springerlink.com/ ***Reprinted with permission. No further reproduction is authorized without written permission from Springer Verlag. This version of the document is not the version of record. Figures and/or pictures may be missing from this format of the document.*** Abstract: Let Com t , q denote the variety of finite monoids that satisfy the equations xy = yx and x t = x t+q . The variety Com 1,1 is the variety of finite semilattices also denoted by J 1 . In this paper, we consider the product variety J 1 *Com t,q generated by all semidirect products of the form M * N with M  J 1 and N  Com t,q . We give a complete sequence of equations for J 1 * Com t,q implying complete sequences of equations for J 1 * (Com A), J 1 * (Com G) and J 1 * Com, where Com denotes the variety of finite commutative monoids, A the variety of finite aperiodic monoids and G the variety of finite groups. Article: 1. Introduction Let Com t , q denote the variety of finite monoids that satisfy the equations xy = yx and x t = x t+q . The variety Com 1,1 is the variety of finite semilattices also denoted by J 1 . In this paper, we give an equational characterization of the product variety J 1 * Com t , q generated by all semidirect products of the form M * N with M  J 1 and N  Com t , q . Our results imply a complete sequence of equations for J 1 * (Com A), J 1 *(Com G) and J 1 *Com, where Com denotes the variety of finite commutative monoids, A the variety of finite aperiodic monoids and G the variety of finite groups. Pin [12] has shown that the variety J 1 * Com 1 , 1 is defined by the equations xux = xux 2 and xuyvxy = xuyvyx. Irastorza [7] has given equations of the particular products J 1 * Com 0,q and has shown that, although the two varieties J 1 and Com 0 , 2 are defined by finite sequences of equations, their product is not. Almeida [1] has given an equational characterization of the variety of finite monoids generated by all semidirect products of i finite semilattices and has shown that it is defined by a finite sequence of equations if and only if i = 1 or 2. Ash [2] has shown that the variety J 1 * G = Inv is defined by the equation x w y w = y w x w , that is, J 1 * G is the variety generated by the inverse semigroups. Our results follow from versions of techniques used in particular by Blanchet-Sadri [3], Brzozowski and Simon [4] and Pin [11, 12]. 1.1. Definitions and notations Let M and N be monoids. We say that M divides N and write M  N if M is a morphic image of a submonoid of N . Note that the divisibility relation is transitive. An M-variety V is a family of finite monoids that satisfies the following two conditions:  If N  V and M  N , then M  V .  If M, N  V , then M  N  V.