Down closed injectivity and essentialness

2020 ◽  
pp. 2150118
Author(s):  
Leila Shahbaz ◽  
Mojgan Mahmoudi
Keyword(s):  
Explicit Description ◽  
Homological Characterization ◽  
Important Notion ◽  
Injective Hulls ◽  
Essential Extensions

Injectivity is one of the useful notions in algebra, as well as in many other branches of mathematics, and the study of injectivity with respect to different classes of monomorphisms is crucial in many categories. Also, essentiality is an important notion closely related to injectivity. Down closed monomorphisms and injectivity with respect to these monomorphisms, so-called dc-injectivity, were first introduced and studied by the authors for [Formula: see text]-posets, posets with an action of a pomonoid [Formula: see text] on them. They gave a criterion for dc-injectivity and studied such injectivity for [Formula: see text] itself, and for its poideals. In this paper, we give results about dc-injectivity of [Formula: see text]-posets, also we find some homological characterization of pomonoids and pogroups by dc-injectivity. In particular, we give a characterization of pomonoids over which dc-injectivity is equivalent to having a zero top element. Also, introducing the notion of [Formula: see text]-injectivity for [Formula: see text]-posets, where [Formula: see text] and [Formula: see text] is externally adjoined to the posemigroup [Formula: see text], we find some classes of pomonoids such that for [Formula: see text]-posets over them the Baer Criterion holds. Further, several kinds of essentiality of down closed monomorphisms of [Formula: see text]-posets, and their relations with each other and with dc-injectivity is studied. It is proved that although these essential extensions are not necessarily equivalent, they behave almost equivalently with respect to dc-injectivity. Finally, we give an explicit description of dc-injective hulls of [Formula: see text]-posets for some classes of pomonoids [Formula: see text].

scholarly journals One-relator groups and algebras related to polyhedral products

2021 ◽  
pp. 1-20
Author(s):  
Jelena Grbić ◽  
George Simmons ◽  
Marina Ilyasova ◽  
Taras Panov
Keyword(s):  
Homology Group ◽  
Homotopy Theory ◽  
Commutator Subgroup ◽  
Homology Groups ◽  
Homological Characterization ◽  
One Relator Groups ◽  
Combinatorial Condition ◽  
The Moment ◽  
Necessary And Sufficient

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

scholarly journals A homological characterization of topological amenability

2012 ◽  
Vol 12 (3) ◽  
pp. 1763-1776
Author(s):  
Jacek Brodzki ◽  
Graham Niblo ◽  
Piotr Nowak ◽  
Nick Wright
Keyword(s):  
Homological Characterization

A Characterization of Almost-Planar Graphs

1996 ◽  
Vol 5 (3) ◽  
pp. 227-245 ◽  
Author(s):  
Bradley S. Gubser
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Planar Graphs ◽  
Explicit Description ◽  
Famous Result ◽  
Minor Criterion ◽  
Excluded Minor

Kuratowski's Theorem, perhaps the most famous result in graph theory, states that K5 and K3,3 are the only non-planar graphs for which both G\e, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the almost-planar graphs, those non-planar graphs for which G\e or G/e is planar for all edges e of G. This paper gives two characterizations of the almost-planar graphs: an explicit description of the structure of almost-planar graphs; and an excluded minor criterion. We also give a best possible bound on the number of edges of an almost-planar graph.

An equivalence criterion for the generalized injectivity of modules with respect to algebraic classes of homomorphisms

2016 ◽  
Vol 15 (09) ◽  
pp. 1650166 ◽  
Author(s):  
Jorge E. Macías-Díaz ◽  
Siegfried Macías
Keyword(s):  
General Definition ◽  
Injective Modules ◽  
Equivalence Criterion ◽  
Definition Of ◽  
Injective Hulls ◽  
Essential Extensions ◽  
The Way

Departing from a general definition of injectivity of modules with respect to suitable algebraic classes of morphisms, we establish conditions under which two modules are isomorphic when they are isomorphic to submodules of each other. The main result of this work extends both Bumby’s criterion for the isomorphism of injective modules and the well-known Cantor–Bernstein–Schröder’s theorem on the cardinality of sets. In the way, various properties on essential extensions, injective modules and injective hulls are generalized. The applicability of our main theorem embraces the cases of [Formula: see text]-injective and pure-injective modules as particular scenarios. Many of the propositions which lead to the proof of the main result of this paper are valid for arbitrary categories.

scholarly journals Characterization of finite amenable transformation semigroups

1973 ◽  
Vol 15 (1) ◽  
pp. 86-93 ◽  
Author(s):  
Carroll Wilde
Keyword(s):  
Transformation Semigroup ◽  
Sufficient Conditions ◽  
Mean Value ◽  
Explicit Description ◽  
Transformation Semigroups ◽  
Shift Operators ◽  
Finite Transformation ◽  
Invariant Means ◽  
Necessary And Sufficient

Abstract. In this paper we develop necessary and sufficient conditions for a finite transformation semigroup to have a mean value which is invariant under the induced shift operators. The structure of such transformation semigroups is described and an explicit description of all possible invariant means given.

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