J. Sichler

On elementary equivalence and isomorphism of clone segments

The principal application of a general theorem proved here shows that for any choice 1≤m≤n≤p of integers there exist metric spacesX andY such that the initialk-segments of their clones of continuous maps coincide exactly whenk≤m, are isomorphic exactly whenk≤n, and are elementarily equivalent exactly whenk≤p.

Almost universal varieties of monoids

The constant mappings onto the unit form a zero subcategory of any category of monoid homomorphisms; a varietyV of monoids isalmost universal if every category of algebras is isomorphic to a class of all nonzero homomorphisms between members ofV. Almost universal monoid varieties are shown to be exactly those varieties containing all commutative monoids in which the identity xnyn=(xy)n fails for every n>1. Almost universal varieties of monoids can also be characterized categorically as the varieties containing all groups with zero as one-object full subcategories.

Equimorphy in varieties of distributive double p-algebras

AbstractAny finitely generated regular variety V of distributive double p-algebras is finitely determined, meaning that for some finite cardinal n(V), any subclass S

Universality of Categories of Coalgebras

\subseteq

Homomorphisms of complete distributive lattices

V of algebras with isomorphic endomorphism monoids has fewer than n(V) pairwise non-isomorphic members. This result follows from our structural characterization of those finitely generated almost regular varieties which are finitely determined. We conjecture that any finitely generated, finitely determined variety of distributive double p-algebras must be almost regular.

Weak automorphisms of universal algebras

Hedrlin and Pultr proved that every small category is isomorphic to a full subcategory of the category Alg (Δ) of all algebras of type Δ whenever the sum ∑Δ of their arities satisfies ∑Δ>2. This article deals with simultaneous representation in categories of algebras, a generalization of the related question: given a subcategory k′ of a small category k, when does there exist an extension Δ′ of a type Δ with ∑Δ≽2 such that k′ is a full subcategory of Alg(Δ′) while the Alg (Δ)-redacts of algebras representing k′ determine a category isomorphic to k?We characterize simultaneous representability by algebras and their reducts completely, and show that it is closely related to Isbells dominion. A consequence of the main result states that algebraically representable pairs (k′, k) of one-object categories k′ k are exactly those for which k′ coincides with its dominion in k, and provides an alternative characterization of the dominion. Simultaneous representability by partial algebras is not subject to any such restriction.

Clones in topology and algebra

Under GCH, a set functor F does not preserve finite unions of non-empty sets if and only if the category Coalg F of all F-coalgebras is universal. Independently of GCH, we show that for any non-accessible functor F preserving intersections, the category Coalg F has a large discrete full subcategory, and we give an example of a category of F-coalgebras that is not universal, yet has a large discrete full subcategory.

Pseudocomplemented distributive lattices with small endomorphism monoids

A quasivariety ℚ is Q-universal if, for any quasivariety \( \mathbb{V} \) of algebraic systems of a finite similarity type, the lattice L(\( \mathbb{V} \)) of all subquasivarieties of \( \mathbb{V} \) is isomorphic to a quotient lattice of a sublattice of the lattice L(ℚ) of all subquasivarieties of ℚ. We investigate Q-universality of finitely generated varieties of distributive double p-algebras. In an earlier paper, we proved that any finitely generated variety of distributive double p-algebras categorically universal modulo a group is also Q-universa1. Here we consider the remaining finitely generated varieties of distributive double p-algebras and state a problem whose solution would complete the description of all Q-universal finitely generated varieties of distributive double p-algebras.