Lattices of homomorphisms

Abstract

Given a variety K of lattice-ordered algebras, A e K is catalytic if for all B e K, K(A, B) is a lattice for the pointwise order. The catalytic objects are determined for various varieties of distributivelattice-ordered algebras. The characterisations obtained do not show an overall unity and exhibit diverse behaviour. Duality is employed extensively. Its usefulness in this context depends on the existence of an order-isomorphism between K(A, B) and the corresponding dual hom-set. Criteria for the existence of such an order-isomorphism are investigated for dualities of the Davey-Werner type. The relationship between catalytic objects and colattices is also discussed.

Equational bases for varieties of Ockham algebras

“…On the contrary, [2], [4], [6] and [8], which are the order duals of [1], [3], [5] and [7] respectively, can be formulated in only one way. The seven non-equivalent g-relations, ordered by implication, are linked as follows: Note also that an element x satisfies [3] if and only if g 3 (x) = g(x).…”

“…Let A = { [1], [2], [3], [4], [6], [8], [9]}. In Table 1, /means that the conjunction of the two axioms does not yield a new axiom of A apart from the obvious ones, × means that one of the two g-relations implies the other and the letter A inside the table indicates that the conjunction implies all the elements of A.…”

“…For instance, let us show that [1] and [6] together imply all the other g-relations. First, [1] and [6] ⇒ [4] since g 2 ≥ g 3 and g 3 ≥ g 0 imply g 2 ≥ g 0 . But [4] and [6] ⇒ [2] since g ≥ g 3 and g 3 ≥ g 0 imply g ≥ g 0 .…”

“…First, [1] and [6] ⇒ [4] since g 2 ≥ g 3 and g 3 ≥ g 0 imply g 2 ≥ g 0 . But [4] and [6] ⇒ [2] since g ≥ g 3 and g 3 ≥ g 0 imply g ≥ g 0 . Hence all the g-relations are satisfied.…”

“…In other words, the axiom (i) is satisfied Vol. 44, 2000 Equational bases for varieties of Ockham algebras 257 (29) = (69) = (9) since [2] as well as [6] implies [9].…”