Multialgebras, power algebras and complete calculi of identities and inclusions

1995 ◽  
pp. 453-468 ◽  
Author(s):  
Michal Walicki ◽  
Sigurd Meldal
Keyword(s):  
Power Algebras

scholarly journals Power Matrices and Dunn--Belnap Semantics: Reflections on a Remark of Graham Priest

2014 ◽  
Vol 11 (1) ◽  
Author(s):  
Lloyd Humberstone
Keyword(s):  
Equational Logic ◽  
Consequence Relations ◽  
Consequence Relation ◽  
Power Matrix ◽  
Matrix Construction ◽  
The Given ◽  
The Universe ◽  
Power Algebras ◽  
Graham Priest ◽  
Natural Way

The plurivalent logics considered in Graham Priest's recent paper of that name can be thought of as logics determined by matrices (in the `logical matrix' sense) whose underlying algebras are power algebras (a.k.a. complex algebras, or `globals'), where the power algebra of a given algebra has as elements \textit{subsets} of the universe of the given algebra, and the power matrix of a given matrix has has the power algebra of the latter's algebra as its underlying algebra, with its designated elements being selected in a natural way on the basis of those of the given matrix. The present discussion stresses the continuity of Priest's work on the question of which matrices determine consequence relations (for propositional logics) which remain unaffected on passage to the consequence relation determined by the power matrix of the given matrix with the corresponding (long-settled) question in equational logic as to which identities holding in an algebra continue to hold in its power algebra. Both questions are sensitive to a decision as to whether or not to include the empty set as an element of the power algebra, and our main focus will be on the contrast, when it is included, between the power matrix semantics (derived from the two-element Boolean matrix) and the four-valued Dunn--Belnap semantics for first-degree entailment a la Anderson and Belnap) in terms of sets of classical values (subsets of {T, F}, that is), in which the empty set figures in a somewhat different way, as Priest had remarked his 1984 study, `Hyper-contradictions', in which what we are calling the power matrix construction first appeared.

Polycharacters of Cocommutative Hopf Algebras

2002 ◽  
Vol 45 (1) ◽  
pp. 11-24 ◽  
Author(s):  
Yuri Bahturin ◽  
Mikhail Kochetov ◽  
Susan Montgomery
Keyword(s):  
Hopf Algebras ◽  
Divided Power ◽  
Enveloping Algebras ◽  
Power Algebras

AbstractIn this paper we extend a well-known theorem of M. Scheunert on skew-symmetric bicharacters of groups to the case of skew-symmetric bicharacters on arbitrary cocommutative Hopf algebras over a field of characteristic not 2. We also classify polycharacters on (restricted) enveloping algebras and bicharacters on divided power algebras.

scholarly journals On the Hopf algebra structure of the Lusztig quantum divided power algebras

2021 ◽  
Vol 27 (2) ◽  
pp. 131-157
Author(s):  
Nicolás Andruskiewitsch ◽  
Iván Angiono ◽  
Cristian Vay
Keyword(s):  
Hopf Algebra ◽  
Algebra Structure ◽  
Divided Power ◽  
Hopf Algebra Structure ◽  
Power Algebras

CATEGORICAL ASPECTS OF POWER ALGEBRAS

1993 ◽  
Vol 16 (2) ◽  
pp. 133-147 ◽  
Author(s):  
H. W. Bargenda ◽  
C. Brink ◽  
V. Vajner
Keyword(s):  
Power Algebras

scholarly journals The module theory of divided power algebras

2017 ◽  
Vol 61 (3-4) ◽  
pp. 287-353
Author(s):  
Rohit Nagpal ◽  
Andrew Snowden
Keyword(s):  
Divided Power ◽  
Module Theory ◽  
Power Algebras

On some congruences of power algebras

2012 ◽  
Vol 10 (3) ◽  
pp. 987-1003 ◽  
Author(s):  
Agata Pilitowska ◽  
Anna Zamojska-Dzienio
Keyword(s):  
Power Algebras

On filtered lie algebras and divided power algebras

1975 ◽  
Vol 3 (7) ◽  
pp. 571-589 ◽  
Author(s):  
Richard C. Block ◽  
Robert Lee Wilson
Keyword(s):  
Lie Algebras ◽  
Divided Power ◽  
Power Algebras

scholarly journals DIVIDED POWER ALGEBRAS OVER AN OPERAD

2019 ◽  
Vol 62 (3) ◽  
pp. 477-517
Author(s):  
SACHA IKONICOFF
Keyword(s):  
Symmetric Group ◽  
The Other ◽  
Divided Power ◽  
Commutative Algebras ◽  
Level Algebras ◽  
Power Algebras

AbstractThe purpose of this paper is to give a characterisation of divided power algebras over a reduced operad. Such a characterisation is given in terms of polynomial operations, following the classical example of divided power algebras. We describe these polynomial operations in two different ways: one way uses invariant elements under the action of the symmetric group and the other coinvariant elements. Our results are then applied to the case of level algebras, which are (non-associative) commutative algebras satisfying the exchange law.

scholarly journals Classification of External Zonotopal Algebras

10.37236/8299 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Gleb Nenashev
Keyword(s):  
Hyperplane Arrangements ◽  
One To One ◽  
Power Algebras

In this paper we work with power algebras associated to hyperplane arrangements. There are three main types of these algebras, namely, external, central, and internal zonotopal algebras. We classify all external algebras up to isomorphism in terms of zonotopes. Also, we prove that unimodular external zonotopal algebras are in one to one correspondence with regular matroids. For the case of central algebras we formulate a conjecture.

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