On the predecessor relation in abstract algebras

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 8, we introduce an algebraic predecessor relation P% and its transitive hull P G coinciding in N with the unary injective successor function ' resp. the <-relation. For some important classes of algebras %, including Peano algebras (absolutely free algebras, word algebras), the algebraic predecessor relation is we.!-founded. Hence, its transitive hull, the natural ordering <% of %, 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 a l l Peano algebras is closed under subalgebras and non-void direct products. -Finally, we will show how in the case of a formal language C, i. e., the Peano algebra C of expressions (=terms & formulas), relations & resp. pr' 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, 03375,03F35, 08A55, 08B20. On the Predecessor Relation in Abstract Algebras 493The notion of an absolutely free algebra emerged first in Mathematical Logic from the syntactical theory of formal languages: The algebra of terms and the algebra of formulas of a formal language, together with the so-called syntactical operations, are absolutely free algebras.There are several wayssetting out from the Dedekind-Peano axiomsto introduce the usual order relation for the natural numbers. One way is the original method ("Kettentheorie") which DEDEKIND himself used in his fundamental booklet [l]. Another one was used by LANDAU in his "Grundlagen der Analysis" [16]. LANDAU first gave an "order-free" proof of a very special recursion principledue to KALMARsufficient to define addition which in turn yields a simple definition of the order relation <, i.e., the transitive hull of the successor function '. In contrast to DEDEKIND'S method the KALM~R-LANDAU method is restricted to the very special case of an algebra with one unary operation and a one-element generating subset.')In this paper we present a detailed treatment of the predecessor relation and its transitive hull in arbitrary algebras. These relations were defined first in L~W I G[18], but solely for the purpose of proving an algebraic recursion principle (homomorphic extension property). L~W I G apparently did not recognize that the axioms characterizing absolutely free algebras generalize the well-known Dedekind-Peano axioms for M. Though he did not use natural numbers when defining the transitive hull of the predecessor relation, he did when proving irreflexivity. Thus his results do not imply DEDEKIND'S. This refers also to L~W I G ' S proof of the algebraic recursion principle, a generalization ...