Model structures, recollements and duality pairs

In this paper, we construct some model structures corresponding Gorenstein [Formula: see text]-modules and relative Gorenstein flat modules associated to duality pairs, Frobenius pairs and cotorsion pairs. By investigating homological properties of Gorenstein [Formula: see text]-modules and some known complete hereditary cotorsion pairs, we describe several types of complexes and obtain some characterizations of Iwanaga–Gorenstein rings. Based on some facts given in this paper, we find new duality pairs and show that [Formula: see text] is covering as well as enveloping and [Formula: see text] is preenveloping under certain conditions, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-injective modules and [Formula: see text] denotes the class of Gorenstein [Formula: see text]-flat modules. We give some recollements via projective cotorsion pair [Formula: see text] cogenerated by a set, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-projective modules. Also, many recollements are immediately displayed through setting specific complete duality pairs.

Duality Pairs and Homomorphisms to Oriented and Unoriented Cycles

In the homomorphism order of digraphs, a duality pair is an ordered pair of digraphs $(G,H)$ such that for any digraph, $D$, $G\to D$ if and only if $D\not \to H$. The directed path on $k+1$ vertices together with the transitive tournament on $k$ vertices is a classic example of a duality pair. In this work, for every undirected cycle $C$ we find an orientation $C_D$ and an oriented path $P_C$, such that $(P_C,C_D)$ is a duality pair. As a consequence we obtain that there is a finite set, $F_C$, such that an undirected graph is homomorphic to $C$, if and only if it admits an $F_C$-free orientation. As a byproduct of the proposed duality pairs, we show that if $T$ is an oriented tree of height at most $3$, one can choose a dual of $T$ of linear size with respect to the size of $T$.

Duality pairs induced by Auslander and Bass classes

Let R and S be arbitrary rings and let C S R {{}_{R}C_{S}} be a semidualizing bimodule, and let 𝒜 C ⁢ ( R op ) {\mathcal{A}_{C}(R^{\mathrm{op}})} and ℬ C ⁢ ( R ) {\mathcal{B}_{C}(R)} be the Auslander and Bass classes, respectively. Then both pairs ( 𝒜 C ⁢ ( R op ) , ℬ C ⁢ ( R ) ) and ( ℬ C ⁢ ( R ) , 𝒜 C ⁢ ( R op ) ) (\mathcal{A}_{C}(R^{\mathrm{op}}),\mathcal{B}_{C}(R))\quad\text{and}\quad(% \mathcal{B}_{C}(R),\mathcal{A}_{C}(R^{\mathrm{op}})) are coproduct-closed and product-closed duality pairs and both 𝒜 C ⁢ ( R op ) {\mathcal{A}_{C}(R^{\mathrm{op}})} and ℬ C ⁢ ( R ) {\mathcal{B}_{C}(R)} are covering and preenveloping; in particular, the former duality pair is perfect. Moreover, if ℬ C ⁢ ( R ) {\mathcal{B}_{C}(R)} is enveloping in Mod ⁡ R {\operatorname{Mod}R} , then 𝒜 C ⁢ ( S ) {\mathcal{A}_{C}(S)} is enveloping in Mod ⁡ S {\operatorname{Mod}S} . Also, some applications to the Auslander projective dimension of modules are given.

Motives in creating an LGBTQ inclusive work environment: a case study

PurposeThe purpose of this inquiry is to examine why companies create LGBTQ-inclusive work environments and how these firms advance LGBTQ-inclusiveness through CSR practices and address challenges presented by strategic duality.Design/methodology/approachUsing a qualitative multiple case study design, data was collected and then triangulated from interviews and company documents. NVivo, a qualitative research program, was used to organize, sort, query and model the data.FindingsSeveral themes were identified as reasons why Fortune 500 organizations sought to create LGBTQ-inclusive work environments. Themes include a positive return on investment, advancing human rights issues within the framework of corporate social responsibility (CSR), internal organizational pressure and parity with other Minnesota companies. Findings are examined through the theoretical lens of strategic duality.Research limitations/implicationsPrimary theoretical implications include contributions to our understanding of strategic duality by providing a first-hand account from people in organizations that encountered imperatives that to some degree are in conflict. By design, the multiple case study methodology does not allow generalizations to be drawn beyond the organizations included in this study.Practical implicationsBoth managers and researchers will find this study provides valuable insight on how people and organizations experience and navigate strategic duality (pairs of competing imperatives) within the context of the motivation behind creating an LGBTQ-inclusive work environment.Originality/valueThis inquiry provides a unique and valuable account as to why organizations choose to invest resources in creating a LGBTQ-inclusive work environment, the return on investment (ROI) and examines competing imperatives (strategic dualities) faced by management. Similar multiple case studies of this qualitative nature are rare, possibly even non-existent and, therefore, this study makes a significant contribution to the literature.

Relative Gorenstein rings and duality pairs

Let [Formula: see text] be a ring (not necessarily commutative) and [Formula: see text] a bi-complete duality pair. We investigate the notions of (flat-typed) [Formula: see text]-Gorenstein rings, which unify Iwanaga–Gorenstein rings, Ding–Chen rings, AC-Gorenstein rings and Gorenstein [Formula: see text]-coherent rings. Using an abelian model category approach, we show that [Formula: see text] and [Formula: see text], the homotopy categories of all exact complexes of projective and injective [Formula: see text]-modules, are triangulated equivalent whenever [Formula: see text] is a flat-typed [Formula: see text]-Gorenstein ring.