77 research outputs found
A Weyl Entropy of Pure Spacetime Regions
We focus on the Penrose's Weyl Curvature Hypothesis in a general framework
encompassing many specific models discussed in literature. We introduce a
candidate density for the Weyl entropy in pure spacetime perfect fluid regions
and show that it is monotonically increasing in time under very general
assumptions. Then we consider the behavior of the Weyl entropy of compact
regions, which is shown to be monotone in time as well under suitable
hypotheses, and also maximal in correspondence with vacuum static metrics. The
minimal entropy case is discussed too
Set-theoretic solutions to the Yang-Baxter equation and generalized semi-braces
This paper aims to introduce a construction technique of set-theoretic
solutions of the Yang-Baxter equation, called strong semilattice of solutions.
This technique, inspired by the strong semilattice of semigroups, allows one to
obtain new solutions. In particular, this method turns out to be useful to
provide non-bijective solutions of finite order. It is well-known braces, skew
braces and semi-braces are closely linked with solutions. Hence, we introduce a
generalization of the algebraic structure of semi-braces based on this new
construction technique of solutions
Set-theoretical solutions of the Yang-Baxter and pentagon equations on semigroups
The Yang-Baxter and pentagon equations are two well-known equations of
Mathematical Physic. If is a set, a map is said
to be a set theoretical solution of the Yang-Baxter equation if where ,
, and and is the flip map, i.e., the map on given by
. Instead, is called a set-theoretical solution of the
pentagon equation if The main
aim of this work is to display how solutions of the pentagon equation turn out
to be a useful tool to obtain new solutions of the Yang-Baxter equation.
Specifically, we present a new construction of solutions of the Yang-Baxter
equation involving two specific solutions of the pentagon equation. To this
end, we provide a method to obtain solutions of the pentagon equation on the
matched product of two semigroups, that is a semigroup including the classical
Zappa product
Solutions of the Yang-Baxter equation and strong semilattices of skew braces
We prove that any set-theoretic solution of the Yang-Baxter equation
associated to a dual weak brace is a strong semilattice of non-degenerate
bijective solutions. This fact makes use of the description of any dual weak
brace we provide in terms of strong semilattice of skew braces
, with . Additionally, we describe the ideals of
and study its nilpotency by correlating it to that of each skew brace
Inverse semi-braces and the Yang-Baxter equation
The main aim of this paper is to provide set-theoretical solutions of the
Yang-Baxter equation that are not necessarily bijective, among these new
idempotent ones. In the specific, we draw on both to the classical theory of
inverse semigroups and to that of the most recently studied braces, to give a
new research perspective to the open problem of finding solutions. Namely, we
have recourse to a new structure, the inverse semi-brace, that is a triple
with a semigroup and an inverse semigroup
satisfying the relation , for all , where is the inverse of in . In particular, we give several constructions of inverse semi-braces
which allow for obtaining solutions that are different from those until known.Comment: 43 page
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