Categories and General Algebraic Structures with Applications
Volume:5 Issue: 1, Jul 2016

This paper is a chronological survey, with no proofs, of a direction in categorical algebra, which is based on categorical Galois theory and involves generalized central extensions, commutators, and internal groupoids in Barr exact Mal’tsev and more general categories. Galois theory proposes a notion of central extension, and motivates the study of internal groupoids, which is then used as an additional motivation for developing commutator theory. On the other hand, commutator theory suggests: (a) another notion of central extension that turns out to be equivalent to the Galois-theoretic one under surprisingly mild additional conditions; (b) a way to describe internal groupoids in ‘nice’ categories. This is essentially a 20 year story (with only a couple of new observations), from introducing categorical Galois theory in 1984 by the author, to obtaining and publishing final forms of results (a) and (b) in 2004 by M. Gran and by D. Bourn and M. Gran, respectively.

Laan in (Ph.D Thesis, Tartu. 1999) introduced the principal weak form of Condition (P) as Condition (PWP) and gave some characterization of monoids by this condition of their acts. In this paper first we introduce Condition (G-PWP), a generalization of Condition (PWP) of acts over monoids and then will give a characterization of monoids when all right acts satisfy this condition. We also give a characterization of monoids, by comparing this property of their acts with some others. Finally, we give a characterization of monoids coming from some special classes, by this property of their diagonal acts and extend some results on Condition (PWP)
to this condition of acts.

In this paper, we define and study the notion of the real-valued functions on a frame L. We show that F(L), consisting of all frame homomorphisms from the power set of R to a frame L, is an f-ring, as a generalization of all functions from a set X into R. Also, we show that F(L) is isomorphic to a sub-f-ring of R(L), the ring of real-valued continuous functions on L. Furthermore, for every frame L, there exists a Boolean frame B such that F(L) is a sub-f-ring of F(B).

We find a criterion for a morphism of coalgebras over a Barr-exact category to be effective descent and determine (effective) descent morphisms for coalgebras over toposes in some cases. Also, we study some exactness properties of endofunctors of arbitrary categories in connection with natural transformations between them as well as those of functors that these transformations induce between corresponding categories of coalgebras. As a result, we find conditions under which the induced functors preserve natural number objects as well as a criterion for them to be exact. Also this enable us to give a criterion for split epis in a category of coalgebras to be effective descent.

This paper is devoted to the study of products of classes of right S-posets possessing one of the flatness properties and preservation of such properties under products. Specifically, we characterize a pomonoid S over which its nonempty products as right S-posets satisfy some known flatness properties. Generalizing this results, we investigate products of right S-posets satisfying Condition (PWP). Finally, we investigate pomonoids over which products of right S-posets transfer an arbitrary flatness property, projectivity, freeness, and regularity to their components.

Locally extended affine Lie algebras were introduced by Morita and Yoshii in [J. Algebra 301(1) (2006), 59-81] as a natural generalization of extended affine Lie algebras. After that, various generalizations of these Lie algebras have been investigated by others. It is known that a locally extended affine Lie algebra can be recovered from its centerless core, i.e., the ideal generated by weight vectors corresponding to nonisotropic roots modulo its centre. In this paper, in order to realize locally extended affine Lie algebras of type A1
, using the notion of Tits-Kantor-Koecher construction, we construct some Lie algebras which are isomorphic to the centerless cores of these algebras.