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P-varieties - a signature independent characterization of varieties of ordered algebras

Authors :
Stephen L. Bloom
Jesse B. Wright
Source :
Journal of Pure and Applied Algebra. 29(1):13-58
Publication Year :
1983
Publisher :
Elsevier BV, 1983.

Abstract

This paper is concerned mainly with classes (categories) of ordered algebras which in some signature are axiomatizable by a set of inequations between terms (‘varieties’ of ordered algebras) and also classes which are axiomatizable by implications between inequations (‘quasi varieties’ of ordered algebras). For example, if the signature contains a binary operation symbol (for the monoid operation) and a constant symbol (for the identity) the class of ordered monoids M can be axiomatized by a set of inequations (i.e. expressions of the form t ≤ t' . However, if the signature contains only the binary operation symbol, the same class M cannot be so axiomatized (since it is not now closed under subalgebras). Thus, there is a need to find structural, signature independent conditions on a class of ordered algebras which are necessary and sufficient to guarantee the existence of a signature in which the class is axiomatizable by a set of inequations (between terms in this signature). In this paper such conditions are found by utilizing the notion of ‘P-categories’. A P-category C is a category such that each ‘Hom-set’ C ( a,b ) is equipped with a distiguished partial order which is preserved by composition. Aside from proving the characterization theorem, it is also the purpose of the paper to begin the investigation of P-categories.

Details

ISSN :
00224049
Volume :
29
Issue :
1
Database :
OpenAIRE
Journal :
Journal of Pure and Applied Algebra
Accession number :
edsair.doi.dedup.....cbc50e1df8a61337f577c0e02b4a6e64
Full Text :
https://doi.org/10.1016/0022-4049(83)90080-4