On the predecessor relation in abstract algebras
Karl‐Heinz Diener · Mathematical logic quarterly · 1993
Abstract In this paper we generalize the Dedekind theory of order for the natural numbers N to abstract algebras with arbitrarily many finitary or infinitary operations. For any algebra 𝔜, we introduce an algebraic predecessor relation P𝔜 and its transitive hull P*𝔜 coinciding in N with the unary injective successor function' resp. the >‐relation. For some important classes of algebras 𝔜, including Peano algebras (absolutely free algebras, word algebras), the algebraic predecessor relation is well‐founded. Hence, its transitive hull, the natural ordering >𝔜 of 𝔜, is a well‐founded partial order, which turns out to be a convenient device for classifying Peano algebras with respect to the number of operations and their arities. Moreover, the property of well‐foundedness is an efficient tool for giving simple proofs of structure theorems as, e. g., that the class of all Peano algebras is closed under subalgebras and non‐void direct products. ‐ Finally, we will show how in the case of a formal language ℒ︁, i. e., the Peano algebra ℒ︁ of expressions (= terms & formulas), relations Pℒ︁, resp. P*ℒ︁ can be used to define basic syntactical notions as occurences of free and bound variables etc. without any reference to a particular representation (“coding”) of the formal language. MSC: 03B22, 03E30, 03E75, 03F35, 08A55, 08B20.