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