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Interassociates of the Free Commutative Semigroup on n Generators

Authors:	Berit Nilsen Givens Karen Linton Amber Rosin Laurie Dishman

Institution:	(1) Cal Poly Pomona, 3801 W. Temple Ave., Pomona, CA 91768, USA;(2) Cumberland University, One Cumberland Square, Lebanon, TN 37087, USA

Abstract:	The interassociates of the free commutative semigroup on n generators, for n > 1, are identified. For fixed n, let (S, ·) denote this semigroup. We show that every interassociate can be written in the form $(S, \ast_{\bar{k}})$ , depending only on a n-tuple $\bar{k}=( k_1, \ldots, k_n)$ . Next, if $(S, \ast_{\bar{k}})$ and $(S, \ast_{\bar{\ell}})$ are isomorphic interassociates of (S, ·) such that $\phi(x_i) = x_j$ , for x_ii and x_j in the generating set of S, then $k_i = \ell_j$ . Moreover, $(S, \ast_{\bar{k}})\cong(S,\ast_{\bar{\ell}})$ if and only if ${\{k_i\}_{i = 1}^n}$ is a permutation of ${\{\ell_i\}_{i = 1}^n}$ .

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