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## On the hull-kernel and inverse topologies as frames

### Algebra universalis (2013-10-01) 70: 197-212 , October 01, 2013

We consider some frame-theoretic properties of the hull-kernel and the inverse topologies on the set of minimal prime ideals of an algebraic frame with the finite intersection property on its compact elements. Denote by *Alg*_{do} the subcategory of *Frm* consisting of such frames together with dense onto coherent maps. We construct a functor
$${{\sf T} : {\bf Alg}_{\rm do} \rightarrow {\bf Frm}}$$
and a natural transformation
$${\tau : {\sf E} \rightarrow {\sf T}}$$
, where *E* is the inclusion functor from *Alg*_{do} to *Frm*.

## Various disconnectivities of spaces and projectabilities of ℓ-groups

### Algebra universalis (2012-10-01) 68: 91-109 , October 01, 2012

*Arch* denotes the category of archimedean *ℓ*-groups and *ℓ*-homomorphisms. *Tych* denotes the category of Tychonoff spaces with continuous maps, and *α* denotes an infinite cardinal or ∞. This work introduces the concept of an *α*cc-disconnected space and demonstrates that the class of *α*cc-disconnected spaces forms a covering class in *Tych*. On the algebraic side, we introduce the concept of an *α*cc-projectable *ℓ*-group and demonstrate that the class of *α*cc-projectable *ℓ*-groups forms a hull class in *Arch*. In addition, we characterize the *α*cc-projectable objects in *W*—the category of *Arch*-objects with designated weak unit and *ℓ*-homomorphisms that preserve the weak unit—and construct the *α*cc-hull for *G* in *W*. Lastly, we apply our results to negatively answer the question of whether every hull class (resp., covering class) is epireflective (resp., monocoreflective) in the category of *W*-objects with complete *ℓ*-homomorphisms (resp., the category of compact Hausdorff spaces with skeletal maps).

## Canonical formulas for k-potent commutative, integral, residuated lattices

### Algebra universalis (2017-02-01): 1-23 , February 01, 2017

Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Indeed, they provide a uniform and semantic way of axiomatising all extensions of intuitionistic logic and all modal logics above K4. Although the method originally hinged on the relational semantics of those logics, recently it has been completely recast in algebraic terms. In this new perspective, canonical formulas are built from a finite subdirectly irreducible algebra by describing completely the behaviour of some operations and only partially the behaviour of some others. In this paper, we export the machinery of canonical formulas to *substructural logics* by introducing canonical formulas for *k*-potent, commutative, integral, residuated lattices (*k*-*CIRL*). We show that any subvariety of *k*-*CIRL* is axiomatised by canonical formulas. The paper ends with some applications and examples.

## Some results about neat reducts

### Algebra universalis (2010-02-01) 63: 17-36 , February 01, 2010

This is a survey article on the concept of neat reducts. An old venerable idea in algebraic logic, in this paper we show why it is regaining momentum.

## Mal’cev algebras with supernilpotent centralizers

### Algebra universalis (2011-04-01) 65: 193-211 , April 01, 2011

Let *A* be a finite algebra in a congruence permutable variety. We assume that for every subdirectly irreducible homomorphic image of *A* the centralizer of the monolith is *n*-supernilpotent. Then the clone of polynomial functions on *A* is determined by relations of arity |*A*|^{n+1}. As consequences we obtain finite implicit descriptions of the polynomial functions on finite local rings with 1 and on finite groups *G* such that in every subdirectly irreducible quotient of *G* the centralizer of the monolith is a *p*-group.

## Collapsing inverse monoids

### Algebra universalis (2007-06-01) 56: 241-261 , June 01, 2007

### Abstract.

In this paper we investigate a class of inverse transformation monoids constructed from finite lattices, and we describe a necessary and sufficient condition for such a transformation monoid to be collapsing.

## The cardinality of the set of all clones containing a given minimal clone on three elements

### Algebra universalis (2012-12-01) 68: 295-320 , December 01, 2012

All minimal clones on three elements were found by B. Csákány. In this paper, for each minimal clone the cardinality of the set of all clones containing this clone is found.

## The number of slim rectangular lattices

### Algebra universalis (2016-02-01) 75: 33-50 , February 01, 2016

Slim rectangular lattices are special planar semimodular lattices introduced by G. Grätzer and E. Knapp in 2009. They are finite semimodular lattices *L* such that the ordered set Ji *L* of join-irreducible elements of *L* is the cardinal sum of two nontrivial chains. After describing these lattices of a given length *n* by permutations, we determine their number, |SRectL(*n*)|. Besides giving recursive formulas, which are effective up to about *n* = 1000, we also prove that |SRectL(*n*)| is asymptotically (*n* - 2)! ·
$${e^{2}/2}$$
. Similar results for patch lattices, which are special rectangular lattices introduced by G. Czédli and E. T. Schmidt in 2013, and for slim rectangular lattice diagrams are also given.

## Oligomorphic clones

### Algebra universalis (2007-08-01) 57: 109-125 , August 01, 2007

### Abstract.

A permutation group on a countably infinite domain is called *oligomorphic* if it has finitely many orbits of finitary tuples. We define a clone on a countable domain to be *oligomorphic* if its set of permutations forms an oligomorphic permutation group. There is a close relationship to *ω-categorical structures*, i.e., countably infinite structures with a first-order theory that has only one countable model, up to isomorphism. Every locally closed oligomorphic permutation group is the automorphism group of an *ω*-categorical structure, and conversely, the canonical structure of an oligomorphic permutation group is an *ω*-categorical structure that contains all first-order definable relations. There is a similar Galois connection between locally closed oligomorphic clones and *ω*-categorical structures containing all *primitive positive* definable relations.

In this article we generalise some fundamental theorems of universal algebra from clones over a finite domain to oligomorphic clones. First, we define *minimal* oligomorphic clones, and present equivalent characterisations of minimality, and then generalise Rosenberg’s five types classification to minimal oligomorphic clones. We also present a generalisation of the theorem of Baker and Pixley to oligomorphic clones.

## More covers of the Boolean variety of unital ℓ-groups

### Algebra universalis (2013-10-01) 70: 149-162 , October 01, 2013

Within the lattice of varieties of pseudo MV-algebras, the variety
$${\mathcal{B}}$$
of Boolean algebras is the least nontrivial variety. Komori identified all varieties of (commutative) MV-algebras that cover
$${\mathcal{B}}$$
. The authors previously identified all solvable varieties of pseudo MV-algebras that cover
$${\mathcal{B}}$$
. We will show the existence of continuum many nonsolvable varieties of pseudo MV-algebras that cover
$${\mathcal{B}}$$
, show that periodically primitive u*ℓ*-groups cannot generate Boolean covers, and show that all noncommutative varieties that are Boolean covers must be Top Boolean.