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A subsemigroup of the rook monoid


We define a subsemigroup \(S_n\) of the rook monoid \(R_n\) and investigate its properties. To do this, we represent the nonzero elements of \(S_n\) (which are \(n\times n\) matrices) via certain triplets of integers, and develop a closed-form expression representing the product of two elements; these tools facilitate straightforward deductions of a great number of properties. For example, we show that \(S_n\) consists solely of idempotents and nilpotents, find the numbers of idempotents and nilpotents, compute nilpotency indexes, determine Green’s relations and ideals, and come up with a minimal generating set. Furthermore, we give a necessary and sufficient condition for the jth root of a nonzero element to exist in \(S_n\), show that existence implies uniqueness, and compute the said root explicitly. We also point to several combinatorial aspects; describe a number of subsemigroups of \(S_n\) (some of which are familiar from previous studies); and, using rook n-diagrams, graphically interpret many of our results.

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We thank the reviewer, whose comments and suggestions notably improved this paper. Theorem 19, in particular, is due to the reviewer. The work of Giannis Fikioris was supported in part by National Science Foundation (NSF) Grant CCF-1408673 and Air Force Office of Scientific Research (AFOSR) grant FA9550-19-1-0183.

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Correspondence to George Fikioris.

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Communicated by Mark V. Lawson.

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Fikioris, G., Fikioris, G. A subsemigroup of the rook monoid. Semigroup Forum 105, 191–216 (2022).

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  • Rook monoid
  • Symmetric inverse semigroup
  • Rook n-diagrams
  • Inverse semigroups
  • Orthodox semigroups
  • Combinatorial semigroups