Sharp partial closure operator. (English) Zbl 1463.06013

Šešelja, Branimir; Slivková, Anna; Tepavčević, Andreja

Miskolc Math. Notes 19, No. 1, 569-579 (2018).

Summary: As an improvement of existing relationships among collections of sets, closure operators and posets, a particular, so called sharp partial closure operator (SPCO) is introduced. It is proved that there is always a unique SPCO corresponding to a given partial closure system. Moreover, an SPCO has the greatest domain among all partial operators corresponding to a given system. If it is a function, an SCPO is a classical closure operator. Dealing with partial closure systems, we introduce principal ones, corresponding to principal ideals of a poset and accordingly, we define principal SPCO’s. Finally, we prove a representation theorem for posets in terms of principal SPCO’s and principal partial closure systems.

MSC:

06A15	Galois correspondences, closure operators (in relation to ordered sets)
06A06	Partial orders, general

Keywords:

partial closure operator; partial closure system; centralized system