P-varieties - a signature independent characterization of varieties of ordered algebras
Journal of Pure and Applied Algebra(1983)
摘要
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.
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