Valdis Laan

PULLBACKS AND FLATNESS PROPERTIES OF ACTS. II

Let S be a monoid. A right S-act A is pullback flat (= strongly flat) if the functor A ⊗–(from the category of left S-acts to the category of sets) preserves pullbacks. We investigate possible generalizations of this notion, obtained either by restricting attention to certain types of pullbacks or by weakening the requirement of pullback preservation. We note that it is possible to describe the already familiar notions of flatness, (principal) weak flatness, and torsion freeness in these terms. Furthermore, a number of new properties arise.

On homological classification of pomonoids by regular weak injectivity properties of S-posets

If S is a partially ordered monoid then a right S-poset is a poset A on which S acts from the right in such a way that the action is compatible both with the order of S and A. By regular weak injectivity properties we mean injectivity properties with respect to all regular monomorphisms (not all monomorphisms) from different types of right ideals of S to S. We give an alternative description of such properties which uses systems of equations. Using these properties we prove several so-called homological classification results which generalize the corresponding results for (unordered) acts over (unordered) monoids proved by Victoria Gould in the 1980’s.

LIMIT PRESERVATION PROPERTIES OF THE GREATEST SEMILATTICE IMAGE FUNCTOR

We study what kinds of limits are preserved by the greatest semilattice image functor from the category of all semigroups to its subcategory of all semilattices.

Context equivalence of semigroups

Two semigroups are called strongly Morita equivalent if they are contained in a Morita context with unitary bi-acts and surjective mappings. We consider the notion of context equivalence which is obtained from the notion of strong Morita equivalence by dropping the requirement of unitariness. We show that context equivalence is an equivalence relation on the class of factorisable semigroups and describe factorisable semigroups that are context equivalent to monoids or groups, and semigroups with weak local units that are context equivalent to inverse semigroups, orthodox semigroups or semilattices.

Generators in the category of S-posets

The paper contains characterizations of generators and cyclic projective generators in the category of ordered right acts over an ordered monoid.

On Mono- and Epimorphisms in Varieties of Ordered Algebras

We study morphisms in varieties of ordered universal algebras. We prove that (i) monomorphisms are precisely the injective homomorphisms and that (ii) every regular monomorphism is an order embedding, but the converse is not true in general. We also give a necessary and sufficient condition for a morphism to be a regular epimorphism. Finally, we discuss factorizations in such varieties.

Principal Weak Flatness and Regularity of Diagonal Acts

For a monoid S, the set S × S equipped with the componentwise right S-action is called the diagonal act of S and is denoted by D(S). A monoid S is a left PP (left PSF) monoid if every principal left ideal of S is projective (strongly flat). We shall call a monoid S left P(P) if all principal left ideals of S satisfy condition (P). We shall call a monoid S weakly left P(P) monoid if the equalities as = bs, xb = yb in S imply the existence of r ∈ S such that xar = yar, rs = s. In this article, we prove that a monoid S is left PSF if and only if S is (weakly) left P(P) and D(S) is principally weakly flat. We provide examples showing that the implications left PSF ⇒ left P(P) ⇒ weakly left P(P) are strict. Finally, we investigate regularity of diagonal acts D(S), and we prove that for a right PP monoid S the diagonal act D(S) is regular if and only if every finite product of regular acts is regular. Furthermore, we prove that for a full transformation monoid S = 𝒯 X , D(S) is regular.

Hamiltonian ordered algebras and congruence extension

Abstract We introduce the notion of Hamiltonian ordered algebra and explore its relationships with various congruence extension properties for ordered algebras. Furthermore, we give sufficient conditions for extending order-congruences or compatible quasiorders from subalgebras.

On Morita Equivalence of Partially Ordered Monoids

We show that there is one-to-one correspondence between certain algebraically and categorically defined subobjects, congruences and admissible preorders of S-posets. Using preservation properties of Pos-equivalence functors between Pos-categories we deduce that if S and T are Morita equivalent partially ordered monoids and F:PosS→PosT is a Pos-equivalence functor then an S-poset AS and the T-poset F(AS) have isomorphic lattices of (regular, downwards closed) subobjects, congruences and admissible preorders. We also prove that if AS has some flatness property then F(AS) has the same property.

Tensor products and preservation of weighted limits, for S-posets

We prove that the functor of tensor multiplication by a right S-poset (S is a pomonoid) preserves all small weighted limits if and only if this S-poset is cyclic and projective. We also show that this functor preserves all finite pie-limits if and only if the S-poset is a filtered colimit of S-posets isomorphic to S S .