Commutative composition semigroups of polynomials. (English) Zbl 0706.08003

Eigenthaler, Günther; Winkler, Reinhard

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.]

Reviewer: Günther Eigenthaler

MSC:

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

Keywords:

composition of polynomials; commutative semigroup; polynomial algebra; unitary left R-modules; commutative monoids; bounded distributive lattices; Boolean algebras