Near sets are disjoint sets that resemble each other. Resemblance is determined by considering set descriptions defined by feature vectors (n-dimensional vectors of numerical features that represent characteristics of objects such as digital image pixels). Near sets are useful in solving problems based on human perception [44,76, 49,51,56] that arise in areas such as image analysis [52,14,41, 48,17,18], image processing [41], face recognition [13], ethology [63], as well as engineering and science problems [53,63,44,19,17,18].As an illustration of the degree of nearness between two sets, consider an example of the Henry color model for varying degrees of nearness between sets [17, §4.3].

Given the binary operation • δ : 2 V ×2 V −→ 2 V defined on a subset of 2 V in the proximal Banach space V , we prove that the monoid S = 2 V (• δ ) is regular, every right ideal A ⊂ S and left ideal B ⊂ S are proximal, every ideal in S is idempotent and S is simple. ForMathematics Subject Classification: 20M32, 52A21, 54E05

The focus of this special issue of Mathematics in Computer Science is near set theory and applications. 1 The study of various forms of nearness relations in proximity space theory and the penultimate notions of near and far in topology 2 span over 100 years, starting with an address by F. Riesz at the 1908 ICM congress in Rome. 3 Basically, two types of near sets are represented in this issue, namely, spatially near sets and, more recently, descriptively near sets. Classical, spatially near sets (nonempty sets are near provided their closures have a nonempty intersection in some compactification) were introduced by F. Herrlich (1974), and others, continuing to the present. 5 Descriptively near sets (pairs of either disjoint or non-disjoint nonempty sets that resemble each other) were inspired by correspondence between J.F. Peters and Z. Pawlak 6 in 2002, collaboration between J.F. Peters, 1 An overview of the theory and applications of proximity space-based near sets is given in

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