Ideal Structure of the Kauffman and Related Monoids

Lau, Kwok Wai; FitzGerald, D. G.

doi:10.1080/00927870600651414

Cited by 43 publications

(32 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, we define the Jones monoid (also called Temperly-Lieb monoid) 2 J n as the submonoid of B n consisting of all diagrams that can be drawn in the plane within a rectangle in such a way that any two of its strings have empty intersection. It is well known and easy to see that J n is aperiodic [18].…”

Section: Partition Monoidsmentioning

confidence: 98%

“…Figure 19 shows the diagrams of these elements of J 2n -here only the 'relevant' part is shown: all remaining strings are 'identity' strings of the form The projections of these elements are shown in Figure 20. According to [18], i˛i 2i 2i 2ǐ…”

Section: Monoids Of Order Preserving Mappings Of a Finite Chainmentioning

confidence: 99%

“…Most papers concerned with these objects investigate the corresponding (twisted) associative algebras and systematic research on the underlying pure monoid structure has been started only since the late 1990's by various authors (see e.g. [9,[17][18][19]). While that research so far has been focussing mainly on combinatorics and classical themes (presentations, Green's relations, ideal structure and the like), it is the intention of the present paper to cross the bridge to modern semigroup and monoid theory as it was established a few decades ago by Eilenberg and Schützenberger [10,11], the Rhodes school [4] and others.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Pseudovarieties generated by Brauer type monoids

Auinger¹

2014

Forum Mathematicum

View full text Add to dashboard Cite

It is proved that the series of all Brauer monoids B n generates the pseudovariety of all finite monoids while the series of their aperiodic analogues, the Jones monoids J n (also called Temperly-Lieb monoids), generates the pseudovariety of all finite aperiodic monoids. The proof is based on the analysis of wreath product decomposition and Krohn-Rhodes theory. The fact that the Jones monoids J n form a generating series for the pseudovariety of all finite aperiodic monoids can be viewed as solution of an old problem popularized by J.-É. Pin. For the latter, the relationship between the Jones monoids J n and the monoids O n of order preserving mappings of a chain of length n is investigated.

show abstract

Section: Partition Monoidsmentioning

confidence: 98%

Section: Monoids Of Order Preserving Mappings Of a Finite Chainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Pseudovarieties generated by Brauer type monoids

Auinger¹

2014

Forum Mathematicum

View full text Add to dashboard Cite

show abstract

“…The set PP n of all planar partitions is clearly a submonoid of P n . The Jones and Motzkin monoids are defined by J n = B n ∩ PP n and M n = PB n ∩ PP n ; see [20,23]. It is well known [19,22] that PP n is isomorphic to J 2n ; see for example [19, p873].…”

Section: Partition Monoidsmentioning

confidence: 99%

Congruence lattices of finite diagram monoids

East

Mitchell

Ruškuc

et al. 2018

Advances in Mathematics

View full text Add to dashboard Cite

We give a complete description of the congruence lattices of the following finite diagram monoids: the partition monoid, the planar partition monoid, the Brauer monoid, the Jones monoid (also known as the Temperley-Lieb monoid), the Motzkin monoid, and the partial Brauer monoid. All the congruences under discussion arise as special instances of a new construction, involving an ideal I, a retraction I → M onto the minimal ideal, a congruence on M , and a normal subgroup of a maximal subgroup outside I.Roughly speaking, if α ∈ P n , then α * is obtained by reflecting (a graph representing) α in the horizontal axis midway between the two rows of vertices. It is easy to see that α * * = α, (αβ) * = β * α * , αα * α = α, for all α, β ∈ P n . It follows that P n is a regular * -semigroup, in the sense of Nordahl and Scheiblich [35], with respect to this operation. We have several obvious identities, such as dom(α * ) = codom(α) and ker(α * ) = coker(α). This symmetry/duality will allow us to shorten several proofs. Other diagram monoidsIn this subsection, we introduce a number of important submonoids of the partition monoid P n . Following [33], the Brauer and partial Brauer monoid are defined by B n = {α ∈ P n : all blocks of α have size 2} and PB n = {α ∈ P n : all blocks of α have size at most 2}, Green's equivalences R, L , J , H and D reflect the ideal structure of a semigroup S, and are the fundamental structural tool in semigroup theory. They are defined as follows. We write S 1 = S if S is a monoid; otherwise S 1 is the monoid obtained from S by adjoining an identity element to S. Then, for a, b ∈ S,further, H = R ∩ L , and D is the join R ∨ L : i.e., the least equivalence containing R and L . It is well known that D = R • L = L • R for any semigroup S, and that D = J when S is finite (as is the case for all semigroups considered in this article). If K is any of Green's relations, and if a ∈ S, we write K a = {b ∈ S : a K b} for the K -class of a in S. The set S/J = {J a : a ∈ S} of all J -classes of S is partially ordered as follows. For a, b ∈ S, we say that J a ≤ J b if a ∈ S 1 bS 1 . If T is a subset of S that is a union of J -classes, we write T /J for the set of all J -classes of S contained in T . The reader is referred to [7, Chapter 2], [21, Chapter 2] or [36, Appendix A] for a more detailed introduction to Green's relations.Green's equivalences on all diagram monoids considered in this article are governed by (co)domains, (co)kernels and ranks, as specified in the following proposition. For P n this was first proved in [40], though the terminology there was different. For the other monoids see [10, Theorem 2.4] and also [16][17][18]40]. The proposition will be used frequently throughout the paper without explicit reference.Proposition 2.1. Let K n be any of the monoids P n , PB n , B n , PP n , M n , I n , J n , O n . If α, β ∈ K n , then (i) α R β ⇔ dom(α) = dom(β) and ker(α) = ker(β), (ii) α L β ⇔ codom(α) = codom(β) and coker(α) = coker(β),Remark 2.2. A number of consequences and simplifications ar...

show abstract

“…So the difference with equivalential Frobenius monads is that here symmetry is missing. The name of Jones monads is derived from the connection of these monads with the monoid J ω of [7] (named with the initial of Jones' name); this monoid is closely related to monoids introduced in [17] (p. 13), which are called Jones monoids in [22] (as suggested by [7]). It can be shown that the category J of the Jones monad freely generated by a single object is isomorphic to a subcategory of Gen.…”

Section: Remark On Jones Monadsmentioning

confidence: 99%

Syntax for split preorders

Došen¹,

Petric²

2013

Annals of Pure and Applied Logic

View full text Add to dashboard Cite

A split preorder is a preordering relation on the disjoint union of two sets, which function as source and target when one composes split preorders. The paper presents by generators and equations the category SplPre, whose arrows are the split preorders on the disjoint union of two finite ordinals. The same is done for the subcategory Gen of SplPre, whose arrows are equivalence relations, and for the category Rel, whose arrows are the binary relations between finite ordinals, and which has an isomorphic image within SplPre by a map that preserves composition, but not identity arrows. It was shown previously that SplPre and Gen have an isomorphic representation in Rel in the style of Brauer.The syntactical presentation of Gen and Rel in this paper exhibits the particular Frobenius algebra structure of Gen and the particular bialgebraic structure of Rel, the latter structure being built upon the former structure in SplPre. This points towards algebraic modelling of various categories motivated by logic, and related categories, for which one can establish coherence with respect to Rel and Gen. It also sheds light on the relationship between the notions of Frobenius algebra and bialgebra. The completeness of the syntactical presentations is proved via normal forms, with the normal form for SplPre and Gen being in some sense orthogonal to the composition-free, i.e. cut-free, normal form for Rel. The paper ends by showing that the assumptions for the algebraic structures of SplPre, Gen and Rel cannot be extended with new equations without falling into triviality.

show abstract

Ideal Structure of the Kauffman and Related Monoids

Cited by 43 publications

References 10 publications

Pseudovarieties generated by Brauer type monoids

Pseudovarieties generated by Brauer type monoids

Congruence lattices of finite diagram monoids

Syntax for split preorders

Contact Info

Product

Resources

About