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Commutative composition semigroups of polynomials. (English) Zbl 0706.08003
General algebra, Dedicated Mem. of Wilfried Nöbauer, Contrib. Gen. Algebra 6, 89-101 (1988).
Let $$\mathcal V$$ be a variety of universal algebras, $$A$$ an algebra of $$\mathcal V$$, $$X=\{x\}$$ a one-element set, and $$A(X,\mathcal V)$$ the $$\mathcal V$$-polynomial algebra in $$X$$ over $$A$$ in the sense of Lausch-Nöbauer [H. Lausch and W. Nöbauer, Algebra of polynomials. Amsterdam etc.: North-Holland (1973; Zbl 0283.12101)]. The elements of $$A(X,\mathcal V)$$ can be represented as $$\mathcal V$$-terms $$p(a_1,\ldots, a_n,x)$$, where $$a_1,\ldots, a_n\in A$$. The composition $$\circ$$, which is defined by substitution of terms:
$p(a_1,\ldots, a_n,x)\circ q(b_1,\ldots, b_m),x):=p(a_1,\ldots, a_n,q(b_1,\ldots, b_mx)),$
is an associative binary operation on $$A(X,\mathcal V)$$.
The paper treats the problem of determining all maximal commutative subsemigroups of the semigroup $$(A(X,\mathcal V),\circ)$$ for the following varieties $$\mathcal V$$: a) unitary left $$R$$-modules $$(R$$ being a ring with identity); b) commutative monoids; c) bounded distributive lattices; d) Boolean algebras. Complete solutions are given in case a) for $$R$$ being a principal ideal domain and $$A$$ being divisible, in case c) for any $$A$$, and in case d) for $$A$$ being atomic.
[For the entire collection see Zbl 0694.00002.]

##### MSC:
 08A40 Operations and polynomials in algebraic structures, primal algebras 13C13 Other special types of modules and ideals in commutative rings 06E30 Boolean functions