### 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 |

### Fingerprint

### 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

*Publicationes Mathematicae*,

*93*(1-2), 215-240. https://doi.org/10.5486/PMD.2018.8232

**Support theorems in abstract settings.** / Olbryś, Andrzej; Páles, Z.

Research output: Contribution to journal › Article

*Publicationes Mathematicae*, vol. 93, no. 1-2, pp. 215-240. https://doi.org/10.5486/PMD.2018.8232

}

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 -