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The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial semantics from final semantics, using the initiality and finality to ensure their equality. Moreover,...
Properties of self maps of BE-algebras are studied. Characterizations of self-distributive BE-algebra, commutative BE-algebras and implicative BE-algebras are derived with the help of left and right self maps. Filters of BE-algebras are characterized with the help of right maps. A set of equivalent conditions is derived for a self map to become bijective.
The critical point between varieties and of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of but of no member of , if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors...
We prove that every distributive algebraic lattice with at most ℵ1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The ℵ1 bound is optimal, as we find a distributive algebraic lattice D with ℵ2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence neither...
In the semantics of programming, finite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with infinite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property...
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