The theory of global fuzzy neighborhood structures (II). Fuzzy topogenous orders

On -proximities of the internal type

“…The initial L-proximities of the internal type, in the categorical sense [1], are the L-proximities of the internal type which provide an initial lift in the category L-PRI. Here, we show that the characterization, introduced in [9,14], for the coarsest L-topogenous structures provides in case of internally symmetric L-topogenous structures (that is, L-proximities of the internal type) an initial lift in L-PRI. This means these coarsest internally symmetric L-topogenous structures are the initial L-proximities of the internal type.…”

“…Most of the results of L-topogenous structures mentioned in this section are given in [14] and reformulated in [9]. Let L be a completely distributive complete lattice with different least and greatest elements 0 and 1, respectively.…”

“…The notion of L-topogenous structure on a set X is defined by Katsaras and Petalas in [14] as a binary relation on the set L X of all L-subsets of X and reformulated by the author and others in [9] as a mapping of L X into the power set P L X of L X . The symmetry property of L-topogenous structure in sense of Katsaras and Petalas, which is called in [9] complementary symmetry, is defined by means of a fixed order-reversing involution of L. The L-proximity in sense of Katsaras [13] is identified with a complementarily symmetric L-topogenous structure.…”

“…The symmetry property of L-topogenous structure in sense of Katsaras and Petalas, which is called in [9] complementary symmetry, is defined by means of a fixed order-reversing involution of L. The L-proximity in sense of Katsaras [13] is identified with a complementarily symmetric L-topogenous structure. In [9] we introduced another notion of symmetry of L-topogenous structure, called internal symmetry, which does not depend on an order-reversing involution. The internally symmetric L-topogenous structure is called L-proximity of the internal type.…”

“…We may note that, in this paper, we use the term "L" instead of the term "fuzzy" (in case of fuzzy topogenous structures, fuzzy proximities of the internal type, global fuzzy neighborhood structures and fuzzy uniform structures) which is used in [8][9][10].…”