2006 Student Research Conference:

19th Annual Student Research Conference

19th Annual Student Research Conference

**The Lattice of Subvarieties of HSP(S+)**

Matthew J. Sealy

Mr. Pamela J. Ryan, Faculty Mentor

The variety generated by a class of algebras is an algebraic structure containing that class which is closed under direct products, subalgebras, and homomorphic images. Our focus is the class of complex algebras of semigroups. The complex algebra of a semigroup is its power set endowed with set union, intersection, complementation, and a binary complex operator. Let S+ denote the class of all complex algebras of semigroups. Then HSP(S+) is the variety generated by S+. Under the familiar set operations, the subvarieties of HSP(S+) form a lattice. In our research, the lattice of subvarieties of HSP(S+) is completed up to varieties generated by four-element Boolean algebras with one normal, associative binary operator and complex algebras of three-element semigroups. In doing so, the collection of all non-isomorphic three-element semigroups is enumerated. In the future, we hope to complete the lattice for all eight-element Boolean algebras with one additional binary operator.

**Keywords:** Mathematics, Universal Algebra, Lattice, Ordered Set, Semigroup, Boolean Algebra

**Topic(s):**Mathematics

**Presentation Type:** Poster

**Session:** 51-4

**Location:** OP 2111

**Time:** 3:30