Dualities for modal algebras from the point of view of triples

In this paper we show how the theory of monads can be used to deduce in a uniform manner several duality theorems involving categories of relations on one side and categories of algebras with homomorphisms preserving only some operations on the other. Furthermore, we investigate the monoidal structure induced by Cartesian product on the relational side and show that in some cases the corresponding operation on the algebraic side represents bimorphisms

Extending congruences on semigroups

The two main results are: (1) Let S be a semigroup which satisfies the relation a b c d = a c b d abcd = acbd , let A be a subsemigroup of Reg S which is a band of groups and let [ φ ] [\varphi ] be a congruence on A. Then [ φ ] [\varphi ] can be extended to a congruence on S. (2) Let S be a compact topological semigroup which satisfies the relation a b c d = a c b d abcd = acbd , let A be a closed subsemigroup of Reg S and let [ φ ] [\varphi ] be a closed congruence on A such that dim φ ( A ) | H = 0 \dim \,\varphi (A)|\mathcal {H} = 0 . Then [ φ ] [\varphi ] can be extended to a closed congruence on S.</p

The lattice of ideals of a compact semilattice

It is shown that, if L is a compact distributive topological lattice with enough continuous join-preserving maps into I to separate points, then there is a continuous lattice homomorphism from M ( L ) \mathcal {M}(L) , the lattice of M-closed subsets of L, onto L. If J ( L ) J(L) , the set of join-irreducible elements of L, is a compact semilattice then L is iseomorphic with M ( J ( L ) ) \mathcal {M}(J(L)) .</p