Complex algebras of semigroups
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The notion of a Boolean algebra with operators (BAO) was first defined by Jonsson and Tarski in 1951. Since that time, many varieties of BAOs, including modal algebras, closure algebras, monadic algebras, and of course relation algebras, have been studied. With the exception of relation algebras, these varieties each have one unary operator. This paper investigates a variety of BAOs with one binary operator. Begin with a semigroup S = (S,·). The complex algebra of S, denoted S+, is a Boolean algebra whose underlying set is the power set of S with set union, intersection and complementation as the Boolean algebra operations. The multiplication operation defined on the semigroup induces a normal, associative binary operation, '*', on the complex algebra as follows: for all subsets A and B contained in S, A * B= a· b:a ϵ A, b ϵ B . Hence, the complex algebra of a semigroup is a BAO with one normal binary associative operator;Let S+ be the class of all complex algebras of semigroups and consider the variety generated by the class S+, denoted V(S+). The tools required to study this variety are developed, including the duality between BAOs and relational structures as it applies to V(S+). A closure operator is defined which is used to determine homomorphic images of members of V(S+). Theorems on the subdirectly irreducible and simples algebras in V(S+) are proved;Next, the structure of V(S+) is analyzed. The general problem of representing a BAO with one binary, normal, associative operator as a member of V(S+) is discussed. Several examples and theorems concerning representation are presented. It is shown that the quasivariety generated by S+ is strictly contained in V(S+). Lastly, the structure of the lattice of subvarieties of V(S+) is investigated. There are precisely two atoms in this lattice; each atom is generated by a two element algebra. Two infinite chains of varieties exist, with the smallest element in each chain a cover of exactly one atom. This leads to a discussion of certain splitting algebras and conjugate varieties. Finally, equations characterizing some important subvarieties are developed.