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A note on complex algebras of semigroups
 Relational and KleeneAlgebraic Methods in Computer Science: Proc. 7th Int. Sem. Relational Methods in Computer Science and 2nd Int. Workshop Applications of Kleene Algebra
, 2003
"... Abstract. The main result is that the variety generated by complex algebras of (commutative) semigroups is not finitely based. It is shown that this variety coincides with the variety generated by complex algebras of partial (commutative) semigroups. An example is given of an 8element commutative B ..."
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Abstract. The main result is that the variety generated by complex algebras of (commutative) semigroups is not finitely based. It is shown that this variety coincides with the variety generated by complex algebras of partial (commutative) semigroups. An example is given of an 8element commutative Boolean semigroup that is not in this variety, and an analysis of all smaller Boolean semigroups shows that there is no smaller example. However, without associativity the situation is quite different: the variety generated by complex algebras of (commutative) binars is finitely based and is equal to the variety of all Boolean algebras with a (commutative) binary operator. A binar is a set A with a (total) binary operation ·, and in a partial binar this operation is allowed to be partial. We write x · y ∈ A to indicate that the product of x and y exists. A partial semigroup is an associative partial binar, i.e. for all x, y, z ∈ A, if (x · y) · z ∈ A or x · (y · z) ∈ A, then both terms exist and evaluate to the same element of A. Similarly, a commutative partial binar is a binar such that if x · y ∈ A then x · y = y · x ∈ A. Let (P)(C)Bn and (P)(C)Sg denote the class of all (partial) (commutative) groupoids and all (partial) (commutative) semigroups respectively. For A ∈ PBn the complex algebra of A is defined as Cm(A) = 〈P (A), ∪, ∅, ∩, A, \, ·〉, where X · Y = {x · y  x ∈ X, y ∈ Y and x · y exists} is the complex product of X, Y ∈ Cm(A). Algebras of the form Cm(A) are examples of Boolean algebras with a binary operator, i.e., algebras 〈B, ∨, 0, ∧, 1, ¬, ·〉 such that 〈B, ∨, 0, ∧, 1, ¬ 〉 is a Boolean algebra and · is a binary operation that distributes over finite (including empty) joins in each argument. A Boolean semigroup is a Boolean algebra with an associative binary operator. For a class K of algebras, Cm(K) denotes the class of all complex algebras of K, H(K) is the class of all homomorphic images of K, and V(K) is the variety generated by K, i.e., the smallest equationally defined class that contains K. The aim of this note is to contrast the equational theory of Cm((C)Bn) with that of Cm((C)Sg). It turns out that the former is finitely based while the latter is not.