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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. then D • and at least one of the groupoids A • , B • , C • is proximal. Mathematics Subject Classification: 20M32, 52A21, 54E05doi:10.12988/ija.2014.410103 fatcat:gmqrwja2jfhgzncjhe42rvt4wq