Abstract
The large rank of a finite semigroup \(\Gamma \), denoted by \(r_5(\Gamma )\), is the least number \(n\) such that every subset of \(\Gamma \) with \(n\) elements generates \(\Gamma \). Howie and Ribeiro showed that \(r_5(\Gamma ) = |V| + 1\), where \(V\) is a largest proper subsemigroup of \(\Gamma \). This work considers the complementary concept of subsemigroups, called prime subsets, and gives an alternative approach to find the large rank of a finite semigroup. In this connection, the paper provides a shorter proof of Howie and Ribeiro’s result about the large rank of Brandt semigroups. Further, this work obtains the large rank of the semigroup of order-preserving singular selfmaps.
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We would like to express our sincere gratitude to the reviewer for his/her encouragement to prepare Sect. 4 and suggestions which helped us in improving the presentation of the paper.
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Communicated by Mikhail Volkov.
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Jitender Kumar, Krishna, K.V. The large rank of a finite semigroup using prime subsets. Semigroup Forum 89, 403–408 (2014). https://doi.org/10.1007/s00233-014-9577-0
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DOI: https://doi.org/10.1007/s00233-014-9577-0