Renominative logics with extended renomination, equality and predicate complement

A new class of program-oriented logical formalisms is investigated – renominative logics with extended renominations, equality predicates, and predicate complement composition. Composition algebras and languages of such logics are described; their semantic properties are investigated. For these logics, a number of logical consequence relations are proposed and investigated, in particular, the logical consequence relations with undefinedness conditions. Properties of these relations form the semantic basis for further construction of sequent-type calculi for the proposed logics.

Order 3 Elements in G2 and Idempotents in Symmetric Composition Algebras

Order three elements in the exceptional groups of type G2 are classiûed up to conjugation over arbitrary fields. Their centralizers are computed, and the associated classification of idempotents in symmetric composition algebras is obtained. Idempotents have played a key role in the study and classification of these algebras.Over an algebraically closed field, there are two conjugacy classes of order three elements in G2 in characteristic not 3 and four of them in characteristic 3. The centralizers in characteristic 3 fail to be smooth for one of these classes.

Klein and conformal superspaces, split algebras and spinor orbits

We discuss [Formula: see text] Klein and Klein-conformal superspaces in [Formula: see text] space-time dimensions, realizing them in terms of their functor of points over the split composition algebra [Formula: see text]. We exploit the observation that certain split forms of orthogonal groups can be realized in terms of matrix groups over split composition algebras. This leads to a natural interpretation of the sections of the spinor bundle in the critical split dimensions [Formula: see text] and [Formula: see text] as [Formula: see text], [Formula: see text] and [Formula: see text], respectively. Within this approach, we also analyze the non-trivial spinor orbit stratification that is relevant in our construction since it affects the Klein-conformal superspace structure.

Gröbner-Shirshov Basis of Twisted Generic Composition Algebras of Affine Type

In this paper, by using PBW bases for the twisted generic composition algebras of affine type, we prove that the set of the skew-commutator relations of the iso-classes of indecomposable representations forms a minimal Gröbner-Shirshov basis for the twisted generic composition algebras of affine type.