A subsemigroup of the rook monoid

Fikioris, George; Giannis, Fikioris,

doi:10.1007/s00233-022-10302-5

Cited by 2 publications

(14 citation statements)

References 11 publications

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“…In a previous paper [3], we introduced a submonoid M n of IS n and studied its properties. The monoid M n consists of the zero matrix together with those matrices of IS n whose 1s lie on a single diagonal and form an uninterrupted block (i.e., no 0 lies between any two 1s).…”

Section: Introductionmentioning

confidence: 99%

“…, n − 1, with d = 0 being the main diagonal), let k be the row of the northwestern 1, and let m be the row of the southeastern 1. The study of [3] was facilitated by representing the elements of M n as triplets d, k, m ∈ Z 3 (d, k, and m are appropriately restricted), and developing a closed-form expression representing the product of two elements.…”

Section: Introductionmentioning

confidence: 99%

“…This short note is an extension that allows d, k, m ∈ R 3 ; the restrictions on the parameters d, k, m, as well as the product formula, remain unaltered. We thus study a new monoid M n , of which the M n of [3] is a submonoid.…”

Section: Introductionmentioning

confidence: 99%

“…To facilitate comparisons with "the integer case," we maintain much of the notation of [3]; for example, we retain the symbol x T for the semigroup inverse * of x ∈ M n . As in [3], 0 and 1 denote monoid zero and identity, and ideal means two-sided ideal. We use the traditional notations for Green's relations, associated equivalence classes, and principal ideals.…”

Section: Introductionmentioning

confidence: 99%

“…We can use the definitions (1), (3), and (4) to show that M n is a noncommutative monoid, whose identity is 1 = 0, 1, n .…”

Section: Introductionmentioning

confidence: 99%

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An extension to "A subsemigroup of the rook monoid"

Fikioris¹,

Giannis²

2023

Preprint

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In a recent paper, we defined an inverse submonoid Mn of the rook monoid and investigated its properties. That investigation was enabled by representing the nonzero elements of Mn (which are n × n matrices) via certain triplets belonging to Z 3 . In this short note, we allow the aforementioned triplets to belong to R 3 . We thus study a new inverse monoid M n, which is a supermonoid of Mn. We prove that the elements of M n are either idempotent or nilpotent, compute nilpotent indexes, and discuss issues pertaining to jth roots. We also describe the ideals of M n, determine Green's relations, show that M n is a supersemigroup of the Brandt semigroup, and prove that M n has infinite Sierpiński rank. While there are similarities between Mn and M n, there are also essential differences. For example, Mn can be generated by only three elements, all ideals of Mn are principal ideals, and there exist x ∈ Mn that do not possess a square root in Mn; but none of these statements is true in M n.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We can use the definitions (1), (3), and (4) to show that M n is a noncommutative monoid, whose identity is 1 = 0, 1, n .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An extension to "A subsemigroup of the rook monoid"

Fikioris¹,

Giannis²

2023

Preprint

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An extension to “A subsemigroup of the rook monoid”

Fikioris,

Fikioris

2023

Semigroup Forum

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A recent paper studied an inverse submonoid $$M_{n}$$ M n of the rook monoid, by representing the nonzero elements of $$M_{n}$$ M n via certain triplets belonging to $${\mathbb {Z}}^3$$ Z 3 . In this note, we allow the triplets to belong to $${\mathbb {R}}^3$$ R 3 . We thus study a new inverse monoid $$\overline{M}_{n}$$ M ¯ n , which is a supermonoid of $$M_{n}$$ M n . We point out similarities and find essential differences. We show that $$\overline{M}_{n}$$ M ¯ n is a noncommutative, periodic, combinatorial, fundamental, completely semisimple, and strongly $${E}^{*}$$ E ∗ -unitary inverse monoid.

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A subsemigroup of the rook monoid

Cited by 2 publications

References 11 publications

An extension to "A subsemigroup of the rook monoid"

An extension to "A subsemigroup of the rook monoid"

An extension to “A subsemigroup of the rook monoid”

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