4,124 research outputs found
Schreier split extensions of preordered monoids
Properties of preordered monoids are investigated and important subclasses of
such structures are studied. The corresponding full subcategories of the
category of preordered monoids are functorially related between them as well as
with the categories of preordered sets and monoids. Schreier split extensions
are described in the full subcategory of preordered monoids whose preorder is
determined by the corresponding positive cone
Mobi algebra as an abstraction to the unit interval and its comparison to rings
We introduce a new algebraic structure, called mobi algebra, consisting of three constants and one ternary operation. The canonical example of a mobi algebra is the unit interval with the three constants 0, 1, and 1/2 and the ternary operation given by the formula x−yx+yz. We study some of its properties and prove that every unitary ring with one half uniquely determines and is uniquely determined by a mobi algebra with one double. Another algebraic structure, called involutive medial monoid (IMM), is considered to establish the passage between rings and mobi algebras.info:eu-repo/semantics/publishedVersio
Categories vs. groupoids via generalised Mal'tsev properties
We study the difference between internal categories and internal groupoids in
terms of generalised Mal'tsev properties---the weak Mal'tsev property on the
one hand, and -permutability on the other. In the first part of the article
we give conditions on internal categorical structures which detect whether the
surrounding category is naturally Mal'tsev, Mal'tsev or weakly Mal'tsev. We
show that these do not depend on the existence of binary products. In the
second part we focus on varieties of algebras.Comment: 30 pages; final published versio
Further remarks on the "Smith is Huq" condition
We compare the 'Smith is Huq' condition (SH) with three commutator conditions
in semi-abelian categories: first an apparently weaker condition which arose in
joint work with Bourn and turns out to be equivalent with (SH), then an
apparently equivalent condition which takes commutation of non-normal
subobjects into account and turns out to be stronger than (SH). This leads to
the even stronger condition that weighted commutators (in the sense of Gran,
Janelidze and Ursini) are independent of the chosen weight, which is known to
be false for groups but turns out to be true in any two-nilpotent semi-abelian
category.Comment: 13 page
On some categorical-algebraic conditions in S-protomodular categories
In the context of protomodular categories, several additional conditions have
been considered in order to obtain a closer group-like behavior. Among them are
locally algebraic cartesian closedness and algebraic coherence. The recent
notion of S-protomodular category, whose main examples are the category of
monoids and, more generally, categories of monoids with operations and
Jo\'{o}nsson-Tarski varieties, raises a similar question: how to get a
description of S-protomodular categories with a strong monoid-like behavior. In
this paper we consider relative versions of the conditions mentioned above, in
order to exhibit the parallelism with the "absolute" protomodular context and
to obtain a hierarchy among S-protomodular categories
- …