SYLLABUS A range of topics for the course follows. 1. Algebraic theories: universal algebra; equational logic; soundness and completeness; theory translations and constructions. 2. Algebras: initial algebras; induction principle; recursive domain equations; free algebras. 3. Presheaves: cartesian closure; essential geometric morphisms; free cocompletions; Kan extensions; coends. 4. Simply typed...

Directly infinite algebras, those E which have a pair of elements x and y where 1=xy≠yx, are well known to sub-algebra isomorphic M∞(K), the set Z+×Z+-indexed matrices only finitely many nonzero entries. When this is actually an ideal, we may analyze algebra in terms extension some A by that is, short exact sequence K-algebras 0→M∞(K)→E→A→0. The present article characterizes all trivial (split)...

Associated with any [0, 1]-valued propositional logic with a complete algebraic semantics, one can consider algebras of families of fuzzy sets over a classical universe, endowed with the appropriate operations. For the three most important schematic extensions of Hájek’s Basic (Fuzzy) Logic, we investigate the existence and the structure of such algebras of fuzzy sets in the corresponding algeb...

Graded-division algebras are building blocks in the theory of finite-dimensional associative graded by a group G. If G is abelian, they can be described, using loop construction, terms central simple graded-division algebras. On other hand, given finite abelian G, any G-graded-division algebra over field F determined, thanks to result Picco and Platzeck, its class (ordinary) Brauer isomorphism ...

We classify all essential extensions of the form $$ 0 \rightarrow \mathcal{B} \mathcal{D} C(X) where $\mathcal{B}$ is a nonunital, simple, separable, finite, real rank zero, $\mathcal{Z}$-stable $C^\*$-algebra with continuous scale, and $X$ finite CW complex. In fact, we prove that there group isomorphism \mathbf{Ext}(C(X), \mathcal{B}) KK(C(X), \mathcal{M}(\mathcal{B})/\mathcal{B}).

A major part of this paper is devoted to an in-depth study j -operators and their properties. This enables us obtain several results on liftings weak DG modules along simple extensions algebras unify the proofs existing obtained by authors these subjects. Finally, we provide a new characterization (weak) lifting property algebras.