Nilpotent Semigroups Research Articles

On separability of semigroups: application to models of physics and discrete optimization

Nilpotent semigroups are used in various fields of physics to model processes like system decay in quantum mechanics, state transitions in statistical mechanics, and simplifying dynamical systems. They also assist in optimizing control processes. The set of nilpotent matrices forms a semigroup and plays a key role in Jordan decomposition, which has many physical applications. In quantum mechanics, the Jordan form helps analyze linear operators on Hilbert spaces, especially with systems involving eigenvalue multiplicities. In vibration analysis, it aids in studying normal modes of mechanical systems with degenerate frequencies. Additionally, Jordan decomposition simplifies control theory for linear time-invariant systems, stability analysis in dynamical systems, and the behavior of coupled oscillators, as well as solving systems of linear differential equations. Certainly, the semigroup theory has even more applications in theoretical computer science: suffice it to say, for example, that some authors identify semigroups and finite automata. In this paper, we address the problem of P-separability of semigroups with respect to three key predicates: equality-separability, divisibility-separability, and subsemigroup-separability, achieved through homomorphisms into a nilpotent semigroup. We present some significant results that advance the understanding of these concepts.

Commutative nilpotent transformation semigroups

Cameron et al. determined the maximum size of a null subsemigroup of the full transformation semigroup T(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {T}(X)$$\\end{document} on a finite set X and provided a description of the null semigroups that achieve that size. In this paper we extend the results on null semigroups (which are commutative) to commutative nilpotent semigroups. Using a mixture of algebraic and combinatorial techniques, we show that, when X is finite, the maximum order of a commutative nilpotent subsemigroup of T(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {T}(X)$$\\end{document} is equal to the maximum order of a null subsemigroup of T(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {T}(X)$$\\end{document} and we prove that the largest commutative nilpotent subsemigroups of T(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {T}(X)$$\\end{document} are the null semigroups previously characterized by Cameron et al.

Σ -Functions of the Categories of Matrix Representations of Nilpotent Semigroups

Σ -Functions of the Categories of Matrix Representations of Nilpotent Semigroups

Свободные частично коммутативные нильпотентные полугруппы, допускающие внешнепланарные и обобщённые внешнепланарные графы Кэли

Свободные частично коммутативные нильпотентные полугруппы, допускающие внешнепланарные и обобщённые внешнепланарные графы Кэли

Not all nilpotent monoids are finitely related

A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every 4-nilpotent monoid is finitely related. We also give an example of a 5-nilpotent monoid that is not finitely related. To our knowledge, this is the first example of a finitely related semigroup where adjoining an identity yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related.

Σ-functions of categories of matrix representations of nilpotent semigroups

Σ-functions of categories of matrix representations of nilpotent semigroups

A generalized variety problem

Let $${\mathscr {C}}\! om $$ denote the variety of all commutative semigroups. For $$n\ge 1$$ let $${{\mathscr {N}}\! il }_n$$ (respectively $${\mathscr {N}}_n$$ ) denote the variety of all nil (respectively nilpotent) semigroups of index n, and let $${\mathscr {N}}\! il $$ (respectively $${\mathscr {N}}$$ ) denote the generalized variety of all nil (respectively nilpotent) semigroups. For a class $${\mathscr {W}}$$ of semigroups let $${\mathscr {L}}({\mathscr {W}})$$ denote the lattice of all varieties contained in $${\mathscr {W}}$$ , and let $${\mathscr {G}}({\mathscr {W}})$$ denote the lattice of all generalized varieties contained in $${\mathscr {W}}$$ . Almeida has proved that the map $$\alpha :{\mathscr {L}}({\mathscr {N}}\! il \cap {\mathscr {C}}\! om )\cup \{{\mathscr {N}}\! il \cap {\mathscr {C}}\! om \}\rightarrow {\mathscr {G}}({\mathscr {N}}\cap {\mathscr {C}}\! om )$$ given by $${\mathscr {W}}\alpha = {\mathscr {W}}\cap {\mathscr {N}}$$ is an isomorphism, and asked whether the extension $${\alpha }': {\mathscr {L}}({\mathscr {N}}\! il ) \cup \{{\mathscr {N}}\! il \}\rightarrow {\mathscr {G}}({\mathscr {N}})$$ of this map is also an isomorphism. The author has previously shown that $${\alpha }'$$ is not injective, using varieties of nil index 5, and noted that the corresponding map $${\alpha }_n : {\mathscr {L}}({{\mathscr {N}}\! il }_n) \rightarrow {\mathscr {G}}({\mathscr {N}} \cap {{\mathscr {N}}\! il }_n)$$ given by $${\mathscr {W}} {\alpha }_n ={\mathscr {W}}\cap {\mathscr {N}}$$ is therefore not injective for $$n\ge 5$$ . The question of smaller values of n was left open, and in this article it is shown that $${\alpha }_n$$ is not injective for $$n\ge 2$$ , using Leech’s square-free words.

The Intersection Problem for Finite Semigroups

The intersection problem for finite semigroups asks, given a set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. In previous work, it was shown that is problem is [Formula: see text]-complete. We introduce compressibility measures as a useful tool to classify the complexity of the intersection problem for certain classes of finite semigroups. Using this framework, we obtain a new and simple proof that for groups and for commutative semigroups, the problem (as well as the variant where the languages are represented by finite automata) is contained in [Formula: see text]. We uncover certain structural and non-structural properties determining the complexity of the intersection problem for varieties of semigroups containing only trivial submonoids. More specifically, we prove [Formula: see text]-hardness for classes of semigroups having a property called unbounded order and for the class of all nilpotent semigroups of bounded order. On the contrary, we show that bounded order and commutativity imply decidability in poly-logarithmic time on alternating random-access Turing machines with a single alternation. We also establish connections to the monoid variant of the problem.

Free left n-trinilpotent trioids

Loday and Ronco introduced the notion of a trioid and presented the free trioid of rank 1. We introduce the variety of left (right) n-trinilpotent trioids which are analogs of left (right) nilpotent semigroups of rank n considered by Schein, and construct the free left (right) n-trinilpotent trioid.

Nilpotency and strong nilpotency for finite semigroups

AbstractNilpotent semigroups in the sense of Mal’cev are defined by semigroup identities. Finite nilpotent semigroups constitute a pseudovariety, MN, which has finite rank. The semigroup identities that define nilpotent semigroups lead us to define strongly Mal’cev nilpotent semigroups. Finite strongly Mal’cev nilpotent semigroups constitute a non-finite rank pseudovariety, SMN. The pseudovariety SMN is strictly contained in the pseudovariety MN, but all finite nilpotent groups are in SMN. We show that the pseudovariety MN is the intersection of the pseudovariety BGnil with a pseudovariety defined by a κ-identity. We further compare the pseudovarieties MN and SMN with the Mal’cev product 𝖩ⓜ𝖦nill.

Free left n-dinilpotent doppelsemigroups

ABSTRACTA doppelsemigroup is an algebraic system consisting of a set with two binary associative operations satisfying certain equations. Commutative dimonoids in the sense of Loday are examples of doppelsemigroups and two interassociative semigroups give rise to a doppelsemigroup. We introduce left (right) n-dinilpotent doppelsemigroups which are analogs of left (right) nilpotent semigroups of rank n considered by Schein. A free left (right) n-dinilpotent doppelsemigroup is constructed and the least left (right) n-dinilpotent congruence on a free doppelsemigroup is characterized. We also establish that the semigroups of the free left (right) n-dinilpotent doppelsemigroup are isomorphic and the automorphism group of the free left (right) n-dinilpotent doppelsemigroup is isomorphic to the symmetric group.

Commuting graphs of boundedly generated semigroups

Araújo, Kinyon and Konieczny (2011) pose several problems concerning the construction of arbitrary commuting graphs of semigroups.We observe that every star-free graph is the commuting graph of some semigroup. Consequently, we suggest modifications for some of the original problems, generalized to the context of families of semigroups with a bounded number of generators, and pose related problems.We construct monomial semigroups with a bounded number of generators, whose commuting graphs have an arbitrary clique number. In contrast to that, we show that the diameter of the commuting graphs of semigroups in a wider class (containing the class of nilpotent semigroups), is bounded by the minimal number of generators plus two.We also address a problem concerning knit degree.

Free left n-dinilpotent dimonoids

We introduce left (right) n-dinilpotent dimonoids which are analogs of left (right) nilpotent semigroups of rank n considered by B. M. Schein. A free left (right) n-dinilpotent dimonoid is constructed and the least left (right) n-dinilpotent congruence on a free dimonoid is characterized.

Counting Semigroups

This paper proves that almost all finite nilpotent semigroups of index 3 are rigid and noncommutative, and gives estimates of the number of these semigroups. The number of finite almost N 3 semigroups of types B, C, and D is also shown to be negligible relative to the number of nilpotent semigroups of index 3. Corresponding results hold for commutative semigroups.

Finite Nilpotent Semigroups of Small Coclass

The parameter coclass has been used successfully in the study of nilpotent algebraic objects of different kinds. In this article a definition of coclass for nilpotent semigroups is introduced and semigroups of coclass 0, 1, and 2 are classified. Presentations for all such semigroups and formulae for their numbers are obtained. The classification is provided up to isomorphism as well as up to isomorphism or anti-isomorphism. Commutative and self-dual semigroups are identified within the classification.

Rees quotients of numerical semigroups

We introduce a class of finite semigroups obtained by considering Rees quotients of numerical semigroups. Several natural questions concerning this class, as well as particular subclasses obtained by considering some special ideals, are answered while others remain open. We exhibit nice presentations for these semigroups and prove that the Rees quotients by ideals of \mathbb{N} , the positive integers under addition, constitute a set of generators for the pseudovariety of commutative and nilpotent semigroups.

The semigroups of order 9 and their automorphism groups

We report the number of semigroups with 9 elements up to isomorphism or anti-isomorphism to be 52,989,400,714,478 and up to isomorphism to be 105,978,177,936,292. We obtained these results by combining computer search with recently published formulae for the number of nilpotent semigroups of degree 3. We further provide a complete account of the automorphism groups of the semigroups with at most 9 elements. We use this information to deduce that there are 148,195,347,518,186 distinct associative binary operations on an 8-element set and 38,447,365,355,811,944,462 on a 9-element set.

Dead-Beat Functional Observers for Discrete-Time LPV Systems With Unknown Inputs

This technical note deals with functional observers for discrete-time linear parameter varying (LPV) systems. More specifically, dead-beat observers for reconstructing linear combinations of the state are proposed. The system inputs are assumed to be unknown and the observers reduce to functions of the output over a finite number of time-steps. The existence of such observers is proven by resorting to the notion of inverse system together with the concepts of maximal robust invariant subspaces and nilpotent semigroups. A constructive approach to derive the explicit equations of the observer is provided. An example illustrates the efficiency and the computational aspects of the method.

Nilpotent semigroups for the characterization of flat outputs of switched linear and LPV discrete-time systems

This paper addresses the problem of flat output characterization for switched linear systems. The characterization is also extended to LPV systems. The characterization is based on the notion of nilpotent semigroups. A complete corresponding recursive algorithm is provided. It stops after a finite number of steps bounded by the dimension of the system. Illustrative examples, for the respective class of switched linear and LPV systems, highlight the efficiency of the characterization.

A CONSTRUCTION OF COMMUTATIVE NILPOTENT SEMIGROUPS

In this paper, we construct nilpotent semigroups S such that <TEX>$S^n=\{0\}$</TEX>, <TEX>$S^{n-1}{\neq}\{0\}$</TEX> and <TEX>${\Gamma}(S)$</TEX> is a refinement of the star graph <TEX>$K_{1,n-3}$</TEX> with center <TEX>$c$</TEX> together with finitely many or infinitely many end vertices adjacent to <TEX>$c$</TEX>, for each finite positive integer <TEX>$n{\geq}5$</TEX>. We also give counting formulae to calculate the number of the mutually non-isomorphic nilpotent semigroups when <TEX>$n=5$</TEX>, 6 and in finite cases.