It is well known that all subvarieties of the variety of all semigroups are not absolutely closed. So, it is worth to find subvarieties of the variety of all semigroups that are closed in itself or closed in the containing varieties of semigroups. We have gone through this open problem and able to determine that the varieties of right [left] normal bands and left [right] quasinormal bands are closed in the varieties of semigroups defined by the identities axy = xa^ny [axy = ay^nx], axy = x^nay [axy =ayx^n] (n > 1); and axy = ax^nay [axy = ayx^ny] (n > 1), axy = a^nxa^ry [axy = ay^rxy^n] (n, r ∈ N), respectively.
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On Closedness of Right(Left) Normal Bands and Left(Right) Quasinormal Bands
Published: 28 April 2023 by MDPI in The 1st International Online Conference on Mathematics and Applications session Algebra and Geometry with Applications to Related Fields
Keywords: Zigzag equations; Variety; Identity; Special semigroup amalgam; Closed; Band.