Computers and universal algebra: some directions

Complexity of Some Problems in Universal Algebra

In this paper we consider the complexity of several problems involving finite algebraic structures. Given finite universal algebras A and B, these problems ask: (1) Do A and B satisfy precisely the same identities? (2) Do they satisfy the same quasi-identities? and (3) Do A and B have the same set of term operations? In addition to the general case in which we allow arbitrary (finite) algebras,...

Some connections between universal algebra and logics for trees

One of the major open problems in automata and logic is the following: is there an algorithm which inputs a regular tree language and decides if the language can be defined in first-order logic? The goal of this paper is to present this problem and similar ones using the language of universal algebra, highlighting potential connections to the structural theory of finite algebras, including Tame...

Geometric Algebra Computers

Geometric algebra covers a lot of other mathematical systems like vector algebra, complex numbers, Plücker coordinates, quaternions etc. and it is geometrically intuitive to work with. Furthermore there is a lot of potential for optimization and parallelization. In this paper, we investigate computers suitable for geometric algebra algorithms. While these geometric algebra computers are working...

Universal Algebra

Universal Dialgebra: Unifying Universal Algebra and Coalgebra

The concept of dialgebra provides a platform under which universal algebra and coalgebra are unified in one theory. Other examples of dialgebras include universal multialgebras and partial algebras. In the dialgebraic setting, several relationships between common features of these various theories are clarified and, in many cases, rather similar proofs of closely related results are combined to...