Boolean Algebras with Operators
Project Description
A Boolean Algebra is a set B together with 2 operations, + and *, that satisfy certain commutative, associative and distributive laws. As an example, the power set of any set is a Boolean Algebra. Boolean Algebras have been studied for almost 150 years, beginning with G. Boole in the 1850s, and culminating in the famous Stone Representation Theorem in the 1930s, which proves that every Boolean algebra can be thought of as a field of sets. There are also important examples coming from Topology and Logic.
Boolean algebras with operators (BAOs) are Boolean algebras upon which additional functions with special properties, called operators, have been defined. Boolean algebras with one additional unary operator have been studied quite a lot in the last 40 years. Included in this class of algebras (among others) are modal algebras, closure algebras, and monadic algebras. However, not much research has occurred in the area of Boolean algebras with an additional binary operator. An example is the class of complex algebras of semigroups.
There are many interesting equations that accompany this class of BAOs. For example, what do the subalgebras look like? What about homormorphic images and direct products? We will look at varieties in the class of complex algebras of semigroups, and ask questions about equations that classify them. We can also look at the (lattice) structure of sub-varieties of the variety of complex algebras of semigroups.
Boolean algebras with operators (BAOs) are Boolean algebras upon which additional functions with special properties, called operators, have been defined. Boolean algebras with one additional unary operator have been studied quite a lot in the last 40 years. Included in this class of algebras (among others) are modal algebras, closure algebras, and monadic algebras. However, not much research has occurred in the area of Boolean algebras with an additional binary operator. An example is the class of complex algebras of semigroups.
There are many interesting equations that accompany this class of BAOs. For example, what do the subalgebras look like? What about homormorphic images and direct products? We will look at varieties in the class of complex algebras of semigroups, and ask questions about equations that classify them. We can also look at the (lattice) structure of sub-varieties of the variety of complex algebras of semigroups.
Background Needed
Ideally, participants would have some background in Algebraic Structures for this project, and an interest in Logic. A programming class would be useful but not required. We may be writing some programs, probably in Perl, to categorize subalgebras.