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Let ℰ n be the monoid of all orientation-preserving and extensive full transformations on { 1 , ... , n }. In this paper, we compute the rank and the idempotent rank of the ideals of the monoid ℰ n . Moreover, we determine the maximal subsemigroups as well as the maximal subsemibands of the ideals of the monoid ℰ n . Our work extends previous results found in the literature. [ABSTRACT FROM AUTHOR]

Brazil et al. ['Maximal subgroups of infinite symmetric groups', Proc. Lond. Math. Soc. (3) 68 (1) (1994), 77–111] provided a new family of maximal subgroups of the symmetric group $G(X)$ defined on an infinite set X. It is easy to see that, in this case, $G(X)$ contains subsemigroups that are not groups, but nothing is known about nongroup maximal subsemigroups of $G(X)$. We provide infinitely many examples of such semigroups. [ABSTRACT FROM AUTHOR]

We study a submonoid of the well studied monoid P O I n of all order-preserving partial injections on an n-element chain. The set I O F n par of all partial transformations in P O I n which are fence-preserving as well as parity-preserving forms a submonoid of P O I n . We describe Green's relations and ideals of I O F n par . For each ideal of I O F n par , we characterize the maximal subsemigroups. We observe that there are three different types of maximal subsemigroups. [ABSTRACT FROM AUTHOR]

Let POPE n be the monoid of all orientation-preserving and extensive partial transformations on n = { 1 , ⋯ , n } . In this paper, we characterize the structure of the generating sets of POPE n , and prove that each generating set of POPE n contains a minimal idempotent generating set of POPE n . Moreover, the minimal generating sets and minimal idempotent generating sets of POPE n coincide. As applications, we compute the number of distinct minimal (idempotent) generating sets of POPE n , and prove that both the rank and the idempotent rank of the monoid POPE n are equal to n 2 + n + 2 2 . Finally, we determine the maximal subsemigroups as well as the maximal idempotent generated subsemigroups of the monoid POPE n . [ABSTRACT FROM AUTHOR]

A partial transformation α on an n-element chain n = { 1 , ⋯ , n } is called order-preserving if x ≤ y implies x α ≤ y α for all x , y ∈ dom (α) and it is called extensive if x ≤ x α for all x ∈ dom (α) . The set of all partial order-preserving extensive transformations on n forms a semiband POE n . In this paper, we compute the rank and the idempotent rank of the ideals of the monoid POE n . Moreover, we determine the maximal subsemigroups as well as the maximal subsemibands of the ideals of the monoid POE n . Our work extends previous results found in the literature. [ABSTRACT FROM AUTHOR]

A proper subsemigroup of a semigroup is maximal if it is not contained in any other proper subsemigroup. A maximal subsemigroup of a finite semigroup has one of a small number of forms, as described in a paper of Graham, Graham, and Rhodes. Determining which of these forms arise in a given finite semigroup is difficult, and no practical mechanism for doing so appears in the literature. We present an algorithm for computing the maximal subsemigroups of a finite semigroup S given knowledge of the Green's structure of S , and the ability to determine maximal subgroups of certain subgroups of S , namely its group H -classes. In the case of a finite semigroup S represented by a generating set X , in many examples, if it is practical to compute the Green's structure of S from X , then it is also practical to find the maximal subsemigroups of S using the algorithm we present. In such examples, the time taken to determine the Green's structure of S is comparable to that taken to find the maximal subsemigroups. The generating set X for S may consist, for example, of transformations, or partial permutations, of a finite set, or of matrices over a semiring. Algorithms for computing the Green's structure of S from X include the Froidure–Pin Algorithm, and an algorithm of the second author based on the Schreier–Sims algorithm for permutation groups. The worst case complexity of these algorithms is polynomial in | S | , which for, say, transformation semigroups is exponential in the number of points on which they act. Certain aspects of the problem of finding maximal subsemigroups reduce to other well-known computational problems, such as finding all maximal cliques in a graph and computing the maximal subgroups in a group. The algorithm presented comprises two parts. One part relates to computing the maximal subsemigroups of a special class of semigroups, known as Rees 0-matrix semigroups. The other part involves a careful analysis of certain graphs associated to the semigroup S , which, roughly speaking, capture the essential information about the action of S on its J -classes. [ABSTRACT FROM AUTHOR]

We describe and count the maximal subsemigroups of many well-known transformation monoids, and diagram monoids, using a new unified framework that allows the treatment of several classes of monoids simultaneously. The problem of determining the maximal subsemigroups of a finite monoid of transformations has been extensively studied in the literature. To our knowledge, every existing result in the literature is a special case of the approach we present. In particular, our technique can be used to determine the maximal subsemigroups of the full spectrum of monoids of order- or orientation-preserving transformations and partial permutations considered by I. Dimitrova, V. H. Fernandes, and co-authors. We only present details for the transformation monoids whose maximal subsemigroups were not previously known; and for certain diagram monoids, such as the partition, Brauer, Jones, and Motzkin monoids. The technique we present is based on a specialised version of an algorithm for determining the maximal subsemigroups of any finite semigroup, developed by the third and fourth authors, and available in the Semigroups package for GAP, an open source computer algebra system. This allows us to concisely present the descriptions of the maximal subsemigroups, and to clearly see their common features. [ABSTRACT FROM AUTHOR]

In this paper,we study the maximal subsemigroups of several semigroups of order-preserving transformations on the natural numbers and the integers, respectively.We determine all maximal subsemigroups of the monoid of all order-preserving injections on the set of natural numbers as well as on the set of integers. Further, we give all maximal subsemigroups of the monoid of all bijections on the integers. For the monoid of all order-preserving transformations on the natural numbers,we classify also all its maximal subsemigroups, containing a particular set of transformations. [ABSTRACT FROM AUTHOR]

In this paper we describe a portion of the subsemigroup lattice of the full transformation semigroup Ω, which consists of all mappings on the countable infinite set Ω. Gavrilov showed that there are five maximal subsemigroups of Ω containing the symmetric group $\operatorname {Sym}(\varOmega )$. The portion of the subsemigroup lattice of Ω which we describe is that between the intersection of these five maximal subsemigroups and Ω. We prove that there are only 38 subsemigroups in this interval, in contrast to the $2^{2^{\aleph_{0}}}$ subsemigroups between $\operatorname {Sym}(\varOmega )$ and Ω. [ABSTRACT FROM AUTHOR]

A partial transformation α on an n-element chain X n is called order-preserving if x ≤ y implies xα ≤yα for all x, y in the domain of α and it is called extensive if x ≤ xα for all x in the domain of α. The set of all partial order-preserving extensive transformations on X n forms a semiband POE n . We determine the maximal subsemigroups as well as the maximal subsemibands of POE n . [ABSTRACT FROM PUBLISHER]

We characterize the maximal regular subsemigroups of the ideals of the semigroup of all order-preserving transformations as well as of the semigroup of all order-preserving or order-reversing transformations on a finite ordered set. [ABSTRACT FROM AUTHOR]

We study the structure of the ideals of the semigroup IOn of all isotone (order-preserving) partial injections as well as of the semigroup IMn of all monotone (order-preserving or order-reversing) partial injections on an n-element set. The main result is the characterization of the maximal subsemigroups of the ideals of IOn and IMn. [ABSTRACT FROM AUTHOR]

Let Tn be the full transformation semigroup on the n-element set Xn. For an arbitrary integer r such that 2 ≤ r ≤ n-1, we completely describe the maximal subsemigroups of the semigroup K(n, r) = {α ∈ Tn : |im α| ≤ r}. We also formulate the cardinal number of such subsemigroups which is an answer to Problem 46 of Tetrad in 1969, concerning the number of subsemigroups of Tn. [ABSTRACT FROM AUTHOR]

We describe maximal subsemigroups of the finite singular transformation semigroup Sing[SUBn] and obtain their classification completely. We also count number of their maximal subsemigroups, and formulate the cardinal number of such subsemigroups. [ABSTRACT FROM AUTHOR]