On Mono- and Epimorphisms in Varieties of Ordered Algebras

On congruence extension properties for ordered algebras

“…It was shown in [8] that surjective epimorphisms of ordered algebras are not necessarily regular. We, however, have the following remark as an immediate consequence of Corollary 28 and Theorem 32 of [8]. (1) π is a Q-homomorphism;…”

“…With the usual definitions of operations we obtain an ordered -algebra A/θ := A/θ, A/θ , . As in [8], we call such an algebra the regular quotient algebra of A by an order-congruence θ . Such quotients were introduced in [3].…”

“…Because the pointwise order is compatible with the composition of homomorphisms, considering the set of homomorphisms from A to B (for each pair of algebras A and B) as a poset with respect to the pointwise order one may in fact regard these categories as Pos-categories (categories enriched over the category Pos of posets and monotone mappings). We recall from [8] that monomorphisms in the categories of ordered algebras are precisely injective homomorphisms.…”

Dominions, zigzags and epimorphisms for partially ordered semigroups

“…A class of posemigroups is called a variety of posemigroups, for instance cf. [7], if it is closed under taking products (endowed with componentwise order), homomorphic images and subposemigroups. Varieties of pomonoids may be defined similarly.…”

Epimorphisms, dominions and amalgamation in pomonoids

“…While speaking in the sense of category theory, morphisms in this article will always refer to pomonoid homomorphisms (order-embeddings are of course allowed as their special case). Monomorphisms of pomonoids are precisely the injective homomorphisms ( [4] Proposition 22). Also it is easy to see that f : S −→ T is an epimorphism of pomonoids if it is such in the category of monoids, where in the latter case we disregard the orders (and hence monotonicity) and treat S and T as monoids.…”