International Journal of Pure and Applied Mathematics Volume 74 No. 1 2012, 55-119 ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu P A ijpam.eu FILTERS ON POSETS AND GENERALIZATIONS Victor Porton 78640, Shay Agnon 32-29 Ashkelon, ISRAEL e-mail: porton@narod.ru url: http://www.mathematics21.org Abstract: They are studied in details properties of filters on lattices, filters on posets, and certain generalizations thereof. Also it’s done some more gen- eral lattice theory research. There are posed several open problems. Detailed study of filters is required for my ongoing research which will be published as “Algebraic General Topology” series. AMS Subject Classification: 54A20, 06A06, 06B99 Key Words: filters, ideals, lattice of filters, pseudodifference, pseudocomple- ment 1. Preface This article is intended to collect in one document the known properties of filters on posets (and some generalizations thereof, namely “filtrators” defined below). It seems that until now were published no reference on the theory of filters. This text is to fill the gap. This text will also serve as the reference base for my further articles. This text provides a definitive place to refer as to the collection of theorems about filters. Detailed study of filters is required for my ongoing research which will be published as http://www.mathematics21.org/algebraic-general-topology.html (Algebraic General Topology) series. Received: October 11, 2011 c  2012 Academic Publications, Ltd. url: www.acadpubl.eu