260 research outputs found
An observation on n-permutability
We prove that in a regular category all reflexive and transitive relations
are symmetric if and only if every internal category is an internal groupoid.
In particular, these conditions hold when the category is n-permutable for some
n.Comment: 6 page
Radicals of 0-regular algebras
We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive
Star-regularity and regular completions
In this paper we establish a new characterisation of star-regular categories,
using a property of internal reflexive graphs, which is suggested by a recent
result due to O. Ngaha Ngaha and the first author. We show that this property
is, in a suitable sense, invariant under regular completion of a category in
the sense of A. Carboni and E. M. Vitale. Restricting to pointed categories,
where star-regularity becomes normality in the sense of the second author, this
reveals an unusual behaviour of the exactness property of normality (i.e. the
property that regular epimorphisms are normal epimorphisms) compared to other
closely related exactness properties studied in categorical algebra.Comment: 13 page
Ideals and Congruences in L-algebras and Pre-L-algebras
We link the recent theory of -algebras to previous notions of Universal
Algebra and Categorical Algebra concerning subtractive varieties, commutators,
multiplicative lattices, and their spectra. We show that the category of
-algebras is subtractive and normal in the sense of Zurab Janelidze, but
neither the category of -algebras nor that of pre--algebras are Mal'tsev
categories, hence in particular they are not semi-abelian. Therefore
-algebras are a rather peculiar example of an algebraic structure
- …