A quasivariety of algebras of finite type is Q-universal if its lattice of subquasivarieties has, as a homomorphic image of a sublattice, the lattice of subquasivarieties of any quasivariety of algebras of finite type. A sufficient condition for a quasivariety to be Q-universal is given, thereby adding, amongst others, the quasivarieties of de Morgan algebras, Kleene algebras, distributive p-algebras, distributive double p-algebras, Heyting algebras, double Heyting algebras, lattices containing the modular lattice M3, 3 , MV-algebras, and commutative rings with unity to the known Q-universal quasivarieties.

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$Q$-universal quasivarieties of algebras

Any boundable binding category contains a proper class of mutually disjoint copies of itself

A category V is called universal (or binding) if every category of algebras is isomorphic to a full subcategory of V. The main result states that a semigroup variety V is universal if and only if it contains all commutative semigroups and fails the identity x"y" = (xy) for every « > 1. Furthermore, the universality of a semigroup \ riety V is equivalent to the existence in V of a nontrivial semigroup whose endomorphism monoid is trivial, and also to the representability of every monoid as the monoid of all endomorphisms of some semigroup in V. Every universal semigroup variety contains a minimal one with this property while there is no smallest universal semigroup variety. 1980 Mathematics subject classification (Amer. Math. Soc): primary 18 B 15, 20 M 07; secondary 20 M 15, 08 A 35.

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Universal varieties of semigroups

Endomorphisms of any algebra A form a monoid End(A) under their composition. In this paper we investigate the endomorphism monoids of bands, and the question of when End(A) determines A6V within a variety V of bands. An early result on determination by endomorphisms not of algebras but partially ordered sets is due to L.M. Gluskin [7]. It says that two partially ordered sets S and T whose endomorphism semigroups are isomorphic must themselves be either isomorphic or antiisomorphic. Analogous results hold for some small varieties of algebras, for example, for a variety of distributive lattices, see B.M. Schein []9], R. McKenzie and C. Tsinakis [14], and by P. Ribenboim [18]; for distributive p-algebras see [i], for Boolean rings K.D. Magill [13], for semilattices [19], and for Brouwerian semilattices [i0] and [21]. In all of these varieties there exist at most two non-isomorphic algebras with isomorphic endomorphism monoids. B.M. Schein [20] proved that there exist at most four non-isomorphic normal bands with ismorphic endomorphism monoids. Our aim is to generalize the results of B.M. Schein [19,20] to larger

Endomorphism monoids of bands

Large k-free algebras

Primitive classes of algebras with two unary idempotent operations, containing all algebraic categories as full subcategories

Each monoid can be represented as the endomorphism monoid of a complete Heyting algebra. More generally, the category of soberT1-spaces and their open continuous maps, and consequently the category of complete Heyting algebras are universal. On the other hand, it is often impossible to represent monoids using more special (Hausdorff) complete Heyting algebras: for instance, each finite commutative endomorphism monoid of such a Heyting algebra is weakly idempotent.

Endomorphisms of complete heyting algebras

In classical (synthetic) geometry, lines and planes are not sets of points. They are entities in their own right, and the geometry is based on relations between them (and points, the other entities present). It is only in analytic geometry that one starts with a set and imposes on it the geometric structure by defining specific subsets.

Aleš Pultr

Papers

Primitive classes of algebras with two unary idempotent operations, containing all algebraic categories as full subcategories

Endomorphisms of complete heyting algebras

Frames: Topology Without Points