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#### Keywords

Key words and phrases: Quasivariety, quasi-identity, lattice of quasivarieties, coatom. Key words and phrases: Universal Horn class, quasivariety, colour-family, graph, relational structure. Key words: Arithmetic, equational systems, f-rings. Key words: Equivalences between varieties, Morita equivalence of Lawvere theories, varietal generator, Boolean algebras, Post algebras.#### Institution

- Altai State University, Dimitrova 66, 656099 Barnaul, Russia, e-mail: budkin@math.dcn-asu.ru 1 (%)
- Department of Mathematics, University of Bremen, D-28359 Bremen, Germany, e-mail: porst@math.uni-bremen.de 1 (%)
- Dipartimento di Matematica, Via del Capitano 15, 53100 Siena, Italy, e-mail:montagna@mailsrv.unisi.it; sebastiani@mailsrv.unisi.it 1 (%)
- Institute of Mathematics, Siberian Branch of RAS, Prosp. Akad. Koptyuga 4, Novosibirsk, 630090, Russia, e-mail: avk@xfiles.cs.nstu.ru 1 (%)

#### Author

##### ( see all 6)

- Budkin, A. I. 1 (%)
- Gorbunov, Viktor 1 (%)
- Kravchenko, Alexandr 1 (%)
- Montagna, Franco 1 (%)
- Porst, Hans-E. 1 (%)

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## On coatoms in lattices of quasivarieties of algebraic systems

### Algebra Universalis (2001-06-01) 46: 15-24 , June 01, 2001

### Abstract.

It is found the necessary condition for the lattice of quasivarieties has a finite set of coatoms. In particular if a quasivariety is generated by a finitely generated abelian-by-polycyclic-by-finite group or a totally ordered group then it has a finite set of proper maximal subquasivarieties. Also it is proved that the set of quasiverbal congruence relations of a finitely defined universal algebra is closed under any meets.

## Equational fragments of systems for arithmetic

### Algebra Universalis (2001-09-01) 46: 417-441 , September 01, 2001

### Abstract.

We investigate the equational fragments of formal systems for arithmetic by means of the equational theory of f-rings and of their positive cones, starting from the observation that a model of arithmetic is the positive cone of a discretely ordered ring. A consequence of the discreteness of the order is the presence of a discriminator, which allows us to derive many properties of the models of our equational theories. For example, the spectral topology of discrete f-rings is a Stone topology. We also characterize the equational fragment of *Iopen*, and we obtain an equational version of Gödel's First Incompleteness Theorem. Finally, we prove that the lattice of subvarieties of the variety of discrete f-rings is uncountable, and that the lattice of filters of the countably generated distributive free lattice can be embedded into it.

## Equivalence for varieties in general and for $ {\cal BOOL} $ in particular

### Algebra Universalis (2000-08-01) 43: 157-186 , August 01, 2000

### Abstract.

The varieties
$ {\cal W} $
equivalent to a given variety
$ {\cal V} $
are characterized in a purely categorical way. In fact they are described as the models of those Lawvere theories which are Morita equivalent to the Lawvere theory of
$ {\cal V} $
which therefore are characterized first. Along this way the conceptual meanings of the *n*-th matrix power construction of a variety and McKenzie's σ-modification of classes of algebras [22] become transparent. Besides other applications not only the well known equivalences between the varieties
$ {\cal P}_m $
of Post algebras of fixed orders *m* and the variety
$ {\cal BOOL} $
of Boolean algebras are obtained; moreover it can be shown that the varieties
$ {\cal P}_m $
are the only varieties equivalent to
$ {\cal BOOL} $
. The results then are generalized to quasivarieties and more general classes of algebras.

## Antivarieties and colour-families of graphs

### Algebra Universalis (2001-06-01) 46: 43-67 , June 01, 2001

### Abstract.

We suggest an algebraic approach to the study of colour-families of graphs. This approach is based on the notion of a congruence of an arbitrary structure. We prove that every colour-family of graphs is a finitely generated universal Horn class and show that for every colour-family the universal theory is decidable. We study the structure of the lattice of colour-families of graphs and the lattice of antivarieties of graphs. We also consider bases of quasi-identities and bases of anti-identities for colour-families and find certain relations between the existence of bases of a special form and problems in graph theory.