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The primary goal of this paper is to present a unified way to transform thesyntax of a logic system into certain initial algebraic structure so that itcan be studied algebraically. The algebraic structures which one may choose forthis purpose are various clones over a full subcategory of a category. We showthat the syntax of equational logic, lambda calculus and first order logic canbe represented as clones or right algebras of clones over the set of positiveintegers. The semantics is then represented by structures derived from leftalgebras of these clones.

Clone Theory: Its Syntax and Semantics, Applications to Universal Algebra, Lambda Calculus and Algebraic Logic