This paper surveys our current state of knowledge (and ignorance) on the use of hidden sorted algebra as a foundation for the object paradigm. Our main goal is to support equational reasoning about properties of concurrent systems of objects, because of its simple and eecient mechanisation. We show how equational speciications can describe objects, inheritance and modules; our treatment of the ...

A lambda theory satisfies an equation between contexts, where a context is a *-term with some ``holes'' in it, if all the instances of the equation fall within the lambda theory. In the main result of this paper it is shown that the equations (between contexts) valid in every lambda theory have an explicit finite equational axiomatization. The variety of algebras determined by the above equatio...

Roughly speaking, an equational problem is a first order formula whose only predicate symbol is =. We propose some rules for the transformation of equational problems and study their correctness in various models. Then, we give completeness results with respect to some “simple” problems called solved forms. Such completeness results still hold when adding some control which moreover ensures ter...

This paper continues the examination of the structure of pseudocomplemented distributive lattices. First, the Congruence Extension Property is proved. This is then applied to examine properties of the equational classes ¿Sn, — lá«So), which is a complete list of all the equational classes of pseudocomplemented distributive lattices (see Part I). The standard semigroups (i.e., the semigroup gene...

Pattern recognition methods can be divided into four di erent categories: statistical or probabilistic, structural, possibilistic or fuzzy, and neural methods. A formal analysis shows that there is a computational complexity versus representational power trade-o between probabilistic and possibilistic or fuzzy set measures, in general. Furthermore, sigmoidal theory shows that fuzzy set membersh...

In this paper we introduce a finitely axiomatizable equational class of relation type algebras which we call REL. This class includes all relation algebras and is closed under relativization to an arbitrary element. Thus REL contains every relativization of ReU , where ReU is the relation algebra of all binary relations on the set U . The main theorem is the REL Representation Theorem 5.4, in w...

Birkhoff [1935] initiated the general study of algebraic structure. Importantly for our concerns here, his development was from (universal) algebra to (equational) logic. Birkhoff’s starting point was the informal conception of algebra based on familiar concrete examples. Abstracting from these, he introduced the concepts of signature and equational presentation, and thereby formalised what is ...

A class of fuzzy linear programming (FLP) problems with fuzzy coefficients based on fuzzy relations is introduced, the concepts of feasible and (α, β)-maximal and minimal solutions are defined. The class of crisp (classical) LP problems can be embedded into the class of FLP ones. Moreover, for FLP problems a new concept of duality is introduced and the weak and strong duality theorems are deriv...