These notes form Lecture Notes of a short course which I will give at 1st School on Universal Logic in Montreux. They cannot be recommended for self studies because, although all definitions and main ideas are included, there are no proofs and examples. I’m going to provide some of them during my lectures, leaving easy ones as exercises. In the first part we discuss some of the most important n...

Let A be a finite algebra with a majority term. We characterize those algebras categorically equivalent to A. The description is in terms of a derived structure with universe consisting of all subalgebras of A × A, and with operations of composition, converse and intersection. The main theorem is used to get a different sort of characterization of categorical equivalence for algebras generating...

We show that for the reductive Tannaka groups of semisimple holonomic $\mathscr{D}$-modules on abelian varieties, every Weyl group orbit weights their universal cover is realized by a conic Lagrangian cycle cotangent bundle. Applications include weak solution to Schottky problem in genus five, an obstruction existence summands subvarieties varieties and criterion simplicity arising Lie algebras.

We study classes of universal algebras in a variety that are closed under the formation of Cartesian products, homomorphic images and extensions. We prove a Birkhoff type theorem for such axiomatic classes. Let V be a variety of algebras of a fixed type τ and let S be a set of first order sentences in the language of τ of the form (∃x1) . . . (∃xn)((u1 = v1) ∧ · · · ∧ (um = vm)), where u1, v1, ...

lthough fuzzy set theory and  sheaf theory have been developed and studied independently,  ulrich hohle shows that a large part of fuzzy set  theory  is in fact a subfield of sheaf theory. many authors have studied mathematical structures, in particular, algebraic structures, in both  categories of these generalized (multi)sets. using hohle's idea, we show that for a (universal) algebra $a$, th...

We consider finite rectangular algebras of finite type as tree recognizers. The type is represented by a ranked alphabet Σ. We determine the varieties of finite rectangular Σ-algebras and show that they form a Boolean lattice in which the atoms are minimal varieties of finite Σ-algebras consisting of projection algebras. We also describe the corresponding varieties of Σ-tree languages and compa...

We characterize pointed varieties of universal algebras in which (A × B)/A ≈ B, i.e. all product projections are normal epimorphisms. 1. Definition. We will say that a pointed category C has normal projections if every product projection A × B → B in C is a normal epimorphism. Equivalently, for any two objects A and B in such a category C, forming the product A×B and then factoring it by A ≈ A×...

lthough fuzzy set theory and  sheaf theory have been developed and studied independently,  Ulrich Hohle shows that a large part of fuzzy set  theory  is in fact a subfield of sheaf theory. Many authors have studied mathematical structures, in particular, algebraic structures, in both  categories of these generalized (multi)sets. Using Hohle's idea, we show that for a (universal) algebra $A$, th...

The paper gives a sufficient condition formulated in a syntactical form for all codescent morphisms of a variety of universal algebras satisfying the amalgamation property to be effective. This result is further used in proving that all codescent morphisms of quasigroups are effective.