Subalgebras of incidence algebras determined by equivalence relations

The main result of this paper is a necessary and sufficient condition on an equivalence relation R defined on the set of intervals of a locally finite partially ordered set S for the set of R-functions to be a subalgebra of the incidence algebra of S over a field of characteristic zero.     

Lattices of Jordan algebras

Commutative Jordan algebras play a central part in orthogonal models. The generations of these algebras is studied and applied in deriving lattices of such algebras. These lattices constitute the natural framework for deriving new orthogonal models through factor aggregation and disaggregation.     

On clones generated by primitive-positive operations of Tarski's relation algebras

Received November 29, 1996; accepted in final form September 5, 1997.     

Derived equivalence of algebras

The derived equivalence and stable equivalence of algebrasR _A ^m and R _B ^m are studied. It is proved, using the tilting complex, thatR _A ^m andR _B ^m are derived-equivalent whenever algebrasA andB are derived-equivalent.     

Lattices of dominions of universal algebras

We fix a universal algebra A and its subalgebra H. The dominion of H in A (in a class M) is the set of all elements a ∈ A such that any pair of homomorphisms f, g: A → M ∈ M satisfies the following: if f and g coincide on H then f(a) = g(a). In association with every quasivariety, therefore, is a dominion of H in A. Sufficient conditions are specified under which a set of dominions form a lattice. The lattice of dominions is explored for down-semidistributivity. We point out a class of algebras (including groups, rings) such that every quasivariety in this class contains an algebra whose lattice of dominions is anti-isomorphic to a lattice of subquasivarieties of that quasivariety. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 26–45, January–February, 2007.     

Lattices of quasiorders on universal algebras

It is proved that every algebraic lattice is isomorphic to the lattice of quasiorders on a universal algebra. Translated fromAlgebra i Logika, Vol. 34, No. 3, pp. 327-328, May-June, 1995. Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-01520.     

Morita equivalence of dual operator algebras

We consider notions of Morita equivalence appropriate to weak* closed algebras of Hilbert space operators. We obtain new variants, appropriate to the dual algebra setting, of the basic theory of strong Morita equivalence, and new nonselfadjoint analogues of aspects of Rieffel’s W^∗-algebraic Morita equivalence.     

Indecomposability of treed equivalence relations

We define rigorously a “treed” equivalence relation, which, intuitively, is an equivalence relation together with a measurably varying tree structure on each equivalence class. We show, in the nonamenable, ergodic, measure-preserving case, that a treed equivalence relation cannot be stably isomorphic to a direct product of two ergodic equivalence relations.     

Endotypes of partial equivalence relations

We classify all partial equivalence relations according to their endotype with respect to endotopisms and compute the cardinalities of endotopism semigroups (strong endotopism monoids, autotopism groups) of partial equivalence relations on a finite set.

     

Interpolation and the Robinson property for logics not closed under the Boolean operations

A variant of the Robinson property (ROB^*) is introduced. The property ROB^* is equivalent to the usual Robinson property and hence is equivalent also to the interpolation property, in any compact logic closed under the Boolean operations. On the other hand, it will be shown that ROB^* is not always equal to the interpolation property, if a logic is not closed under them. Next, we will study ROB^* and various interpolation properties for equational logics, which are typical examples of compact logics not closed under the Boolean operations. It will be shown that an equational logic has ROB^* if and only if it has the amalgamation property for isomorphic embeddings.Presented by Bjarni Jonsson.     

Morita equivalence of quasi-primal algebras and sheaves

In this paper, we show that two quasi-primal algebras are Morita equivalent if and only if their inverse semigroups of inner automorphisms are isomorphic, and if they have the same one-element subalgebras. The proof of this statement uses the representation theory of algebras by sections in sheaves.Presented by H. P. Gumm.     

Derived equivalence of symmetric special biserial algebras

The Brauer complex of a symmetric SB-algebra is introduced. The so-far known invariants of the stable and derived equivalence of symmetric SB-algebras are stated in terms of the Brauer complex. In particular, the genus of the Brauer complex turns out to be invariant under derived equivalence. Transformations of Brauer complexes that preserve the class of derived equivalence are studied. The SB-algebras with Brauer complex of genus 0 are classified. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 343, 2007, pp. 5–32.