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On the predecessor relation in abstract algebras
Author(s) -
Diener KarlHeinz
Publication year - 1993
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.19930390152
Subject(s) - mathematics , peano axioms , unary operation , finitary , idempotence , partial function , natural number , dedekind cut , binary relation , distributive property , universal algebra , relation algebra , associative property , algebra over a field , discrete mathematics , pure mathematics , algebra representation , two element boolean algebra
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 , we introduce an algebraic predecessor relation P and its transitive hull P * 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 well‐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 all Peano algebras is closed under subalgebras and non‐void direct products. ‐ Finally, we will show how in the case of a formal language ℒ, i. e., the Peano algebra ℒ of expressions (= terms & formulas), relations P ℒ, resp. P * ℒ 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.