A genoid is a category of two objects such that one is the product of itself with the other. A genoid may be viewed as an abstract substitution algebra. It is a remarkable fact that such a simple concept can be applied to present a unified algebraic approach to lambda calculus and first order logic.
The primary goal of this paper is to present a unified way to transform the syntax of a logic system into certain initial algebraic structure so that it can be studied algebraically. The algebraic structures which one may choose for this purpose are various clones over a full subcategory of a category. We show that the syntax of equational logic, lambda… (More)
In this paper, first-order logic is interpreted in the framework of universal algebra, using the clone theory developed in three previous papers   and . We first define the free clone T ÔL, CÕ of terms of a first order language L over a set C of parameters in a standard way. The free right algebra F ÔL, CÕ of formulas over the clone T ÔL, CÕ of… (More)
The concept of a clone is central to many branches of mathematics, such as universal algebra, algebraic logic, and lambda calculus. Abstractly a clone is a category with two objects such that one is a countably infinite power of the other. Left and right algebras over a clone are covariant and contravariant functors from the category to that of sets… (More)
Algebraic logic studies algebraic theories related to proposition and first-order logic. A new algebraic approach to first-order logic is sketched in this paper. We introduce the notion of a quantifier theory, which is a functor from the category of a monad of sets to the category of Boolean algebras, together with a uniquely determined system of… (More)
An abstract machine is a theoretical model designed to perform a rigorous study of computation. Such a model usually consists of configurations, instructions, programs , inputs and outputs for the machine. In this paper we formalize these notions as a very simple algebraic system, called a configuration machine. If an abstract machine is defined as a… (More)