# Support theorems in abstract settings

Andrzej Olbryś, Z. Páles

Research output: Contribution to journalArticle

### 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 215-240 26 Publicationes Mathematicae 93 1-2 https://doi.org/10.5486/PMD.2018.8232 Published - Jan 1 2018

### Fingerprint

Support Theorem
Algebraic Structure
Completeness
Generalized Convex Functions
Extension Theorem
Partially Ordered Set
Interior Point
Convex Sets
Vector space
Deduce
Extremes
Cone
Imply
Sufficient Conditions
Range of data
Family

### Keywords

• 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 journalArticle

Olbryś, Andrzej ; Páles, Z. / Support theorems in abstract settings. In: Publicationes Mathematicae. 2018 ; Vol. 93, No. 1-2. pp. 215-240.
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