Abstract
In this paper, we establish a general framework in which the verification of support theorems for generalized convex functions acting between an algebraic structure and an ordered algebraic structure is still possible. As for the domain space, we allow algebraic structures equipped with families of algebraic operations whose operations are mutually distributive with respect to each other. We introduce several new concepts in such algebraic structures, the notions of convex set, extreme set, and interior point with respect to a given family of operations, furthermore, we describe their most basic and required properties. In the context of the range space, we introduce the notion of completeness of a partially ordered set with respect to the existence of the infimum of lower bounded chains, we also offer several sufficient conditions which imply this property. For instance, the order generated by a sharp cone in a vector space turns out to possess this completeness property. By taking several particular cases, we deduce support and extension theorems in various classical and important settings.
| Original language | English |
|---|---|
| Pages (from-to) | 215-240 |
| Number of pages | 26 |
| Journal | Publicationes Mathematicae |
| Volume | 93 |
| Issue number | 1-2 |
| DOIs | |
| Publication status | Published - Jan 1 2018 |
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Keywords
- Additive controllability
- Algebraic interior
- Convexity
- Delta-convexity
- Extension theorem
- Extreme set
- Generalized affine function
- Generalized convex function
- Generalized convexity
- Lower chain-completeness
- Partially ordered set
- Sandwich theorem
- Sharp cone
- Support theorem
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Support theorems in abstract settings. / Olbryś, Andrzej; Páles, Z.
In: Publicationes Mathematicae, Vol. 93, No. 1-2, 01.01.2018, p. 215-240.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Support theorems in abstract settings
AU - Olbryś, Andrzej
AU - Páles, Z.
PY - 2018/1/1
Y1 - 2018/1/1
N2 - In this paper, we establish a general framework in which the verification of support theorems for generalized convex functions acting between an algebraic structure and an ordered algebraic structure is still possible. As for the domain space, we allow algebraic structures equipped with families of algebraic operations whose operations are mutually distributive with respect to each other. We introduce several new concepts in such algebraic structures, the notions of convex set, extreme set, and interior point with respect to a given family of operations, furthermore, we describe their most basic and required properties. In the context of the range space, we introduce the notion of completeness of a partially ordered set with respect to the existence of the infimum of lower bounded chains, we also offer several sufficient conditions which imply this property. For instance, the order generated by a sharp cone in a vector space turns out to possess this completeness property. By taking several particular cases, we deduce support and extension theorems in various classical and important settings.
AB - In this paper, we establish a general framework in which the verification of support theorems for generalized convex functions acting between an algebraic structure and an ordered algebraic structure is still possible. As for the domain space, we allow algebraic structures equipped with families of algebraic operations whose operations are mutually distributive with respect to each other. We introduce several new concepts in such algebraic structures, the notions of convex set, extreme set, and interior point with respect to a given family of operations, furthermore, we describe their most basic and required properties. In the context of the range space, we introduce the notion of completeness of a partially ordered set with respect to the existence of the infimum of lower bounded chains, we also offer several sufficient conditions which imply this property. For instance, the order generated by a sharp cone in a vector space turns out to possess this completeness property. By taking several particular cases, we deduce support and extension theorems in various classical and important settings.
KW - Additive controllability
KW - Algebraic interior
KW - Convexity
KW - Delta-convexity
KW - Extension theorem
KW - Extreme set
KW - Generalized affine function
KW - Generalized convex function
KW - Generalized convexity
KW - Lower chain-completeness
KW - Partially ordered set
KW - Sandwich theorem
KW - Sharp cone
KW - Support theorem
UR - http://www.scopus.com/inward/record.url?scp=85054567704&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85054567704&partnerID=8YFLogxK
U2 - 10.5486/PMD.2018.8232
DO - 10.5486/PMD.2018.8232
M3 - Article
AN - SCOPUS:85054567704
VL - 93
SP - 215
EP - 240
JO - Publicationes Mathematicae
JF - Publicationes Mathematicae
SN - 0033-3883
IS - 1-2
ER -