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Characterizations of matroid VIA OFR-Sets

The aim of this paper is to introduce the class of OFR-sets as the sets that are the intersection of an open set and a feeble-regular set. Several classes of matroids are studied via the new concept. New decompositions of strong maps are provided. | Turk J Math 25 (2001) , 445 – 455. ¨ ITAK ˙ c TUB Characterizations of Matroid VIA OFR-Sets Talal Ali Al-Hawary Abstract The aim of this paper is to introduce the class of OFR-sets as the sets that are the intersection of an open set and a feeble-regular set. Several classes of matroids are studied via the new concept. New decompositions of strong maps are provided. Key Words: Feeble-matroid, Strong map, Hesitant map, ORF-set, OFF-set, OFRset. 1. Introduction For an introduction on matroids see [3, 4, 7, 8, 9]. In particular, a matroid M is an ordered pair (E ,O) such that O is a collection of subsets, called open sets of M, of a finite set E , called the ground set of M, such that ∅ is an open set, unions of open sets are open and if O1 and O2 are open sets and x ∈ O1 ∩ O2 , there exists an open set O3 such that (O1 ∪ O2 ) − (O1 ∩ O2 ) ⊆ O3 ⊆ (O1 ∪ O2 ) − {x}. An equivalent way of defining a matroid M, is that M is an ordered pair (E , FM ) such that FM is a collection of subsets, called flats or closed sets of M, of a finite set E such that E ∈ FM , intersections of flats are flats and if F ∈ FM and {F1 , F2 , ., Fk } is the set of minimal members of FM (with respect to inclusion) that properly contain F , then ¯ Clearly F ∪ F ∪ . ∪ F = E. The closure of a subset A ⊆ E will be denoted by A. 1 2 k A¯ is the smallest flat containing A and x ∈ A¯ if and only if for every open set O in M that contains x, O ∩ A 6= ∅, see Oxley [4]. A is a spanning set of M if A¯ = E. Let M1 = (E 1 ,F1 ) and M2 = (E 2 ,F2 ) be matroids. A strong map f from M1 to M2 is a map f : E1 → E2 such that the inverse image of any flat of M2 is a flat of M1 . We abbreviate 1994 AMS Subject Classification. 05B35. 445 AL-HAWARY this as f : M1 → M2 . Clearly, f is strong if and only if the inverse image of any open set in M2 is open in M1 . A set U ⊆ E is called a feeble-open set (=FO-set) in M if there ¯ see Al-Hawary [1]. A subset A ⊆ E is exists an open set O ∈ O such that O ⊆ U