Tropical plactic algebra, the cloaktic monoid, and semigroup representations

0.62 MB·

by  Zur Izhakian

TROPICAL PLACTIC ALGEBRA,
THE CLOAKTIC MONOID,
AND SEMIGROUP REPRESENTATIONS
7
1 ZURIZHAKIAN
0
2
Abstract. AnewtropicalplacticalgebraisintroducedinwhichtheKnuthrelationsareinferredfrom

n the underlying semiring arithmetics, encapsulating the ubiquitous plactic monoid Pn. This algebra
a manifests a natural framework for accommodating representations of Pn, or equivalently of Young
J tableaux, and its moderate coarsening — the cloaktic monoid Kn and the co-cloaktic monoid coKn.
8 The faithful linear representations of Kn and coKn by tropical matrices, which constitute a tropical
1 placticalgebra,areshowntoprovidelinearrepresentationsoftheplacticmonoid. Tothisendthepaper
develops a special type of configuration tableaux, corresponding bijectively to semi-standard Young
] tableaux. Thesespecial tableaux allow asystematic encoding ofcombinatorial properties innumerical
O algebraic ways, including algorithmic benefits. The interplay between these algebraic-combinatorial
structuresestablishes aprofoundmachineryforexploringsemigroupattributes, inparticularsatisfying
C
of semigroup identities. This machinery is utilized here to prove that Kn and coKn admit all the
h. semigroupidentities satisfiedbynˆntriangulartropicalmatrices,whichholdsalsoforP3.
t
a
m
[

Contents
1
v Introduction 1
6 1. Preliminaries 5
5
2. Tropical plactic algebra 9
1
3. Tropical matrices 16
5
4. Troplactic matrix algebra 21
0
5. The cloaktic and theco-cloaktic monoids 24
.
1 6. Configuration tableaux 31
0
7. Representations of tableaux and of theplactic monoid 39
7
8. Remarks and open problems 46
1
References 47
:
v AppendixA. Proofs and completions to §7.1 48
i
X
r
a
Introduction
Thispaperintroduceslinearrepresentationsoftheplacticmonoidanditscoarsening,calledthecloaktic
and the co-cloaktic monoids, together with a new encapsulating algebra. The plactic monoid P “
I
PLCpA q is the presented monoid A˚{ . That is, the quotient of the free monoid A˚ over an ordered
I I ”knu I
alphabetA bythecongruence” determinedbytheKnuth relations(alsocalledplacticrelations)
I knu
acb“cab if aďbăc,
(KNT)
bac“bca if aăbďc.
Date:January17,2017.
2010 Mathematics Subject Classification. Primary: 06D05, 06F05, 20M05, 20M13, 20M30, 47D03; Secondary: 05C25,
03D15,16R10,20B30, 16S15,14T05,16Y60.
Key words and phrases. Idempotent semiring, tropical plactic algebra, tropical matrix algebra, colored weighted di-
graph, semigroup identity, forwardsemigroup, plactic monoid, cloaktic monoid, semigroup representation, young tableau,
configurationtableau, symmetricgroup.
TheresearchoftheauthorhasbeensportedbytheResearchCouncilsUK(EPSRC),grantnoEP/N02995X/1.
InstituteofMathematics,UniversityofAberdeen,AB243UE,Aberdeen, UK.
Email: [email protected].
1
2 ZURIZHAKIAN
This monoid was first appeared in the context of Young tableaux (Knuth [26] and Schensted [45]) and
has been followed by an extensive study of Lascoux and Schu¨tzenberger [30], establishing an impor-
tant link between combinatorics and algebra via its bijective correspondence to semi-standard Young
tableaux [25].
The combinatoricsof P is framedby Young tableaux andhas beenstudied mainly in traditionalper-
I
spective (e.g., Gr¨obner-Shirshovbases [4]). It has various applications (e.g., in symmetric functions [37],
Kostka-Foulkes polynomials [31], and Schubert polynomials [32]). Related representations have looked
at monoid algebras KrP s over a field K for a finite I [7, 27]. At the same time, Young tableaux has
I
been foundto be animportantcharacteristicpatterns inclassicalrepresentationtheory ofgroups[8,10],
especially in representations of the symmetric group [44] and algebraic combinatorics [36], as well as in
combinatorics [35] and computer science [41]. Therefore, in this challenging arena, a direct algebraic
description of the plactic monoid strengthens the mutual connections to classical representation theory,
providing an additional algebraic-combinatorialapproach to group representations.
To address this goal, this paper introduces
‚ a semiring structure in which the Knuth relations (KNT) are inferred from its arithmetics,
‚ meaningful coarser monoids of the plactic monoid, and
‚ linear representations of the plactic monoid and its coarsening.
TheseobjectspaveanewwaytostudytheplacticmonoidandYoungtableauxsystematically. Todevelop
linear representation, we are assisted by several auxiliary objects (i.e., semigroups, tropical matrices,
digraphs, and configuration tableaux), enhancing the interplay among them.
The fundamental structure of the monoids of our main interest, determined as a quotient by multi-
plicative congruences,makestheir analysisrather difficult. To approachsuch monoids straightforwardly,
we adopt the familiar concept of associating an algebra to a multiplicative group which allows one to
study groups in terms of algebras, e.g., the correspondence between Lie groups and Lie algebras. This
paper applies a similar concept to the plactic monoid P by encapsulating P in the tropical plactic
I I
algebra plc — a new semiring structure — abbreviated troplactic algebra. The underlying struc-
I
ture of this algebra is a noncommutative idempotent semiring (Defunition 2.4), generated by a set of
orderedelements. In this algebrathe Knuth relations (KNT) areintrinsically followedfromthe semiring
arithmetics (Theorem 2.8).
Animportantpropertyofplc isthateveryproductofits elementsisuniquelydecomposabletoasum
I
of non-nested terms, each is a nondecreasing subsequence (Corollary 2.12). Furthermore, the generators
of plc admit the Frobenius, property (Lemma 2.15):
I
a`b m “am`bm for any mPN.
A dual versionofthe troplactic a`lgebra˘,denoted plc^, exists (Theorem2.20)and is perfectly compatible
I
with the co-mirroring map (CMR) below.
ÝÑ
AÝnÑimmediateconsequenceofthe equivalenceu”knu v oftwowordsinPI isthatthe lengthslenAIpuq
and lenA pvq of longest nondecreasing subsequences (subwords) in u and v are the same. Furthermore,
I
ÝÑ ÝÑ
lenAJpuq“lenAJpvq for every convex sub-alphabet AJ of AI, (CLK)
which happens due to correspondence of P with semi-standard Young tableaux (Proposition 6.9). The
I
converse implication, however, does not hold in general, which leads to defining the cloaktic monoid
K “ CLKpA q as the presented monoid A˚{ , a moderate coarsen of P . That is, A˚{ is the
I I I ”clk I I ”clk
free monoid A˚ subject to the congruence ” , determined by the relations (CLK). Then, the monoid
I clk
homomorphism
ð:P ÝÝÝ։K
I I
is surjective and preserves the congruence ” .
knu
As expected, the homomorphism ð has a critical role in the construction of our linear representations
of the plactic monoid, obtained from those of K . Nevertheless, sometimes, K appeared to be too
I I
coarser, which leads us to pursuing additional representable monoids that respect the congruence ” .
knu
To receive such a monoid, we restrict the frameworkto finitely generatedmonoids A˚ and introduce the
n
co-mirroring of letters
coM pa q:“a a ¨¨¨a a ¨¨¨a , a PA ,
n ℓ n n´1 n´ℓ`2 n´ℓ 1 ℓ n
TROPICAL PLACTIC ALGEBRA 3
which induces the co-mirroring
coM pwq:“ coM pa q¨¨¨ coM pa q, w“a ¨¨¨a , (CMR)
n n ℓ1 n ℓm ℓ1 ℓm
over whole A˚. It also determines the (lexicographic) order preserving surjective homomorphism
n
B :A˚ ãÝÝÝÝÑpA˚qr1s ĂA˚, w ÞÝÑ coM pwq.
1 n n n n
The homomorphism B extends inductively to a chain of endomorphisms B : pA˚qri´1s ãÝÑ pA˚qris of
1 i n n
induced free monoids.
Our co-mirroring construction respects the congruence ” (Theorem 7.2). Thus, for the finitely
knu
generated plactic monoid P , the monoid homomorphism
n
B :P ãÝÝÝÝÑpP qr1s ĂP
1 n n n
isinjective—amonoidembedding. Ontheotherhand, B doesnotpreservetheequivalenceof” with
1 clk
the relations (CLK), but it determines the equivalence ”co, defined by
clk
u”cov ô B puq” B pvq. (CCLK)
clk 1 clk 1
wTihtehctohe-csluorajkecttiicvemmoonnooiidd hcooKmnom“orcpoChiLsKmpAconðq:isPcoÝn։stitcuotKed,apsrocvoKidning“aAs˚nec{o”nccoldk,ctohaartseinsinagccoofmPpa.nied
n n n
The maingoalofthis paper isto constructlinearrepresentationsofthe plactic monoidP interms of
n
tropicalrepresentationsofthe cloakticmonoidK andthe co-cloakticmonoid coK . Therefore,aspecial
n n
focus is given to formulating the correspondences between P to K and coK by relying upon Young
n n n
tableaux.
We start by describing the combinatorialstructure of the cloaktic monoid K in terms of a troplactic
n
algebra plc , obtained by using tropical matrices. Besides their conventional algebraic meanings, these
I
matrices are also combinatorial entities, corresponding uniquely to weighted digraphs. As such, they
intimatelycomposegraphtheoryintropicalalgebraicmethodologies,exhibitingausefulinterplaybetween
algebra and combinatorics. The latter plays a major role in theoretical algebraic studies [2, 21] and in
applications to combinatorics [3, 24], as well as in semigroup representations [15, 19] and automata
theory [47, 48].
This mutual connection allows for producing a special class of tropical matrices that generate the
finite tropical plactic algebra A (Theorem 4.2) along with recording lengths of the longest pathes in
n
digraphs. In turn, these paths encode the longest nondecreasing sequences of the represented elements
(Lemma 4.1). As being a troplactic algebra,the multiplicative submonoidAˆ of A immediately admits
n n
the Knuth relations (KNT). More precisely, Aˆ is isomorphic to the cloaktic monoid K (Theorem 5.6).
n n
Thus, it introduces the faithful tropical linear representation
✵:K ÝÝÝ„ÝÑAˆ.
n n
This isomorphism yields an efficient algorithm for computing the maximal lengths of all nondecreasing
subwords of w P A` with respect to every convex sub-alphabet of A (Algorithm 5.14). Similarly, for
n n
the co-cloaktic monoid coK , we obtain the monoid isomorphism Ω : coK ÝÝ„Ñ˛Aˆ, whose image ˛A
n n n n
is a dual troplactic matrix algebra plc^ (Theorem 5.19). Both isomorphisms ✵ and Ω are allocated with
I
tropical characters that specify characteristic invariants.
Animmediate resultofthe existence ofthe isomorphisms✵ and Ω is thatboth monoids K and coK
n n
admit all the semigroup identities satisfied by the monoid TMat pTq of tropical triangular matrices
n
(Corollaries 5.8 and 5.22). A particular form of these semigroup identities is constructed as
Π : w x w “w y w, (SID)
pC,p,qq
where w :“w is a fixed word over the variables C “tx,yu that contains as factors all the possible
pC,p,qq
words of length q over C in which no letter apperars srequerntialrly more than p times, and such that the
words wxw and wyw also satisfy this law [14, 15]. With this form in place, TMat pTq satisfies the
r r n
identities (SID) with p “ q “ n´1 by letting x “ uv and y “ vu. The identities (SID) generalize the
Adjan’s identity of the bicyclic monoid [1], which is also faithfully represented by tropical matrices [19].
r r r r
Also, TMat pTq satisfies a recursive version of Adjan’s identity [39].
n
With tropical representations of the monoids K and coK in hand, our next goal is to formulate the
n n
surjective monoid homomorphisms ð:P Ý։K and coð:P Ý։ coK explicitly, and to explore their
n n n n
properties as reflected in the representations ✵ : K ÝÝ„ÑAˆ and Ω : coK ÝÝ„Ñ˛Aˆ. To this end, we
n n n n
4 ZURIZHAKIAN
rely upon the well known one-to-one correspondence between the plactic monoid P and semi-standard
n
Young tableaux Tab (e.g., see [29]). In a sense, tableaux are graphical patterns that accommodate
n
symbols, i.e., letters of the associated plactic monoid, with a deep combinatorialmeaning. Nevertheless,
an additional machinery is required to canonically frame Young tableaux by tropical matrices and to
convert visual-combinatorialinformation to suitable numerical-algebraicdata.
Thisconvertingmachineryisprovidedbyn-configurationtableauxCTab . Thatis,tableauxoffixed
n
isoscelestriangularshapethatcontainnon-negativeintegerssubjecttocertainstructurallaws,calledcon-
figuration laws (Definition 6.11). Configuration tableaux correspond bijectively to semi-standard Young
tableaux (Theorem 6.16) and are endowed with a self implementation of the Encoding Algorithm 6.19
that simulates the Bumping Algorithm 6.3 of Tab . Their fixed shape enables the introduction of a
n
canonical reference system, employed to state their correspondence to tropical matrices in Aˆ and ˛Aˆ;
n n
thereby, to link tableaux with weighted digraphs. In this framework, encoding a letter a P A in a
ℓ n
configuration tableaux C P CTab is interpreted as multiplying the matrix ✵pwq by the matrix ✵pa q
w n ℓ
in Aˆ or, dually, Ωpwq by Ωpa q in ˛Aˆ. Accordingly, Tab and CTab are considered as multiplicative
n ℓ n n n
monoidswhoseoperationsareinducedbyletterencoding;hence,ourtableaucorrespondencesarerealized
as monoid homomorphisms.
With all desired components at our disposal, the tropical linear representation
℘ :P ÝÝÝ։Aˆˆ˛Aˆ
n n n n
of the plactic monoid P is obtained by composing the monoid homomorphisms of our main objects
n
(Theorem 7.17), summarized by the diagram
T
ctab
PLC „ //Tab „ //CTab
✻n✻■✻✻■✻■✻✻■✻■✻✻ð■B✻■■■$$C$$ LKnn „✵ Cm//aAt (cid:15)(cid:15)(cid:15)(cid:15)ˆn✿n✿✿✿✿✿✿✿✿✿C✿mcoat
✻ ✿
✻✻ a ✿✿
✻ ✿
✻ ✿
co✻P(cid:27)(cid:27)(cid:27)(cid:27) LC ð //// &&c&&oCLK „Ω //%%%% ✿(cid:28)(cid:28)(cid:28)(cid:28)˛Aˆ .
n 55 n n
coð
℘n $$
Aˆˆ˛Aˆ tt
00 n n
This representation naturally induces a congruence on P , and therefore also an equivalence relation on
n
tableaux in Tab and CTab .
n n
In the case of the plactic monoid of rank 3, P , the semigroup representation ℘ : P ÝÝ„ÑAˆ ˆ
3 3 3 3
˛Aˆ is faithful (Theorem 7.18), more precisely, it is an isomorphism. Consequently, we conclude that
3
P admits all the semigroup identities satisfied by the monoid TMat pTq of 3ˆ3 triangular tropical
3 3
matrices (Corollary 7.19); in particular, the identities Π in (SID). Furthermore, relying upon the
pC,2,2q
correspondencebetweenreversalofwordsandtranspositionoftableaux,standardYoungtableauxSTab
n
areshowntobefaithfullyrealizablebytropicalmatrices(Theorem7.20). AstableauxinSTab bijectively
n
correspond to elements of the symmetric group S , a tropical realization of S is obtained, linking the
n n
theory to Heacke algebras.
Tropical representation theory turns out to be applicable for studying characteristic properties of
semigroups in places that the use of classical representation theory is limited. It provides an alternative
approachforrealizationandforthe explorationofalgebraic-combinatorialobjects;especially,their semi-
group identities. These identities are of special interest in the theory of semigroup varieties [42], e.g.,
in finitely generated semigroups of polynomial growth [11, 46]. Applications of the ubiquitous plactic
monoidin grouprepresentationsare wellknown[10], e.g.,in computing products of Schur functions in n
variables which are the irreducible polynomial characters of the general linear group GL pCq, cf. [34].
n
Tropicallinearrepresentationspavethe waytostudyingcombinatorialaspectsinclassicalrepresentation
theory, including a geometric perspective via projective modules [18, 38]. This paper ease the use of
tropical representation by providing a solid bridge between tropical algebra and classical representation
theory.
TROPICAL PLACTIC ALGEBRA 5
Paper outline.
Forthereaderconvenience,thispaperisdesignedasaself-containeddocument. In§1,webringallthe
relevant definitions and properties of objects to be used in this paper, including basic examples. In §2,
we introduce and study the new structure of tropical plactic algebra and its core object, called forward
semigroup(Definition 2.2). Abriefoverviewontropicalmatricesandtheir associateddigraphsopens§3,
followed by the characterization of tropical corner matrices, providing faithful linear representations of
forward semigroups. In §4 we employ corner matrices to construct an explicit troplactic algebra A
n
and to establish the linkage between diagraphs to nondecreasing subwords. The entire §5 is devoted
to the introduction of the cloaktic monoid K and the co-cloaktic monoid coK , including their linear
n n
representations by A and ˛A , respectively. Young tableaux and configuration tableaux are discussed
n n
in §6, with a special emphasis on numerical functions that allow their mapping to matrices in A .
n
Finally, in §7 we compose our various components to obtain representations and co-representations of
configuration tableaux which eventually result as linear representations of the plactic monoid.
To help for a better understanding, our exposition involves many diagrams and examples, including
of pathologicalcases.
1. Preliminaries
To make this paper reasonably self contained, this section, expect parts of §1.3, recalls the relevant
notions and terminology, as well as definitions and properties of the algebraic structures to be used in
the paper, starting with our special notations.
1.1. Notations.
Unless otherwise is specified, the capital letters I,J denote subsets of the neutral numbers N :“
t1,2,...u, while N stands fort0,1,2,...u. The finite sett1,...,nu is oftendenoted by N, forshort. By
0
“ordered” we always mean totaly ordered.
Definition 1.1. A subset J of an ordered set I is called convex, written J Ď I, if for any i,j P J,
cx
every k PI such that iăkăj, also belongs to J. We write ti:ju for the convex subset of I determined
by iďj.
In other words, a convex subset is an “interval”, could be empty or a singleton. For a given n, we
define i1 :“ n´i`1, for i “ 1,...,n, ordered now reversely as n1 ă pn´1q1 ă ¨¨¨ ă 11. Then tj1 : i1u
has the same number of elements as ti:ju has, and is again convex in N.
WewriteL “rℓ ,ℓ ,...,ℓ sforasequenceofelementsℓ takenfromt1,...,nu,wheret“1,...,m.
m 1 2 m t
This notation means that the indexing order of elements is preserved. We denote the sequence r1,...,ns
by r1:ns, and write ri:js for the subsequence ri,i`1,...,js of r1:ns.
‚ A subsequence S of L is notated as S ĎL , e.g., ri:jsĎr1:ns, for 1ďiďj ďn.
m m
‚ len S denotes the length of S ĎL , i.e., the number of its elements.
m
‚ S denotes a subsequence of length k, k ě0, where S stands for the empty sequence.
k ` ˘ 0
‚ The notation a stands for the product a ¨a ¨¨¨ a , respecting the indexing of the
sPSk s s1 s1 sk
sequence S “rs ,...,s s.
k 1 k
ś
‚ A subsequence S “rs ,s ,...,s s of length k of L is
k 1 2 k m
– non-decreasing,denoted SÒ ĎL , if s ďs 﨨¨ďs ,
k m 1 2 k
– increasing if s ăs 㨨¨ăs ,
1 2 k
– non-increasing, denoted SÓ ĎL , if s ěs 쨨¨ěs ,
k m 1 2 k
– decreasing if s ąs ą¨¨¨ąs .
1 2 k
‚ SÒ and SÓ denote respectively non-decreasing and non-increasing subsequences of an arbitrary
length.
‚ We denote the set of all non-decreasing subsets SÒ ĎL of L by SqÒpL q.
m m m
‚ A non-decreasing subsequence SÒ Ď L is said to be maximal in L if L has no other non-
m m m
decreasing subsequence RÒ Ď L such that SÒ Ĺ RÒ. MSqÒpL q denotes the subset of all
m m
maximal non-decreasing subsequences of L .
m
‚ SqÓpL q and MSqÓpL q are defined similarly for non-increasing subsequences.
m m
6 ZURIZHAKIAN
In this paper we deal only with finite sequences, which are also termed “words” in the context of semi-
groups.
1.2. Free monoids.
AsemigroupS :“pS,¨qisasetofelementstogetherwithanassociativebinaryoperation. Amonoid
is a semigroup with identity element e. Any semigroup S can be formally adjoined with an identity
element e by declaring that ea“ae“a for all aPS, so when dealing with multiplicative structures we
workwithmonoids. Wewriteai fora¨a¨¨¨awithaPS repeateditimes,andformallyidentifya0 withe,
when S is a monoid. An element o of S is said to be absorbing, usually identified as 0, if oa “ ao“o
for all aPS. S is a pointed semigroup if it has an absorbing element o.
An Abelian semigroup S is cancellative with respect to a subset T Ď S if ac “ bc implies a “ b
whenever a,b P S and c P T. In this case, T is called a cancellative subset of S, and it generates a
subsemigroupinS, alsocancellative. Thus,weusually assumethatT is asubsemigroup. AsemigroupS
is strictly cancellative if S is cancellative with respect to itself. The term “congruence” refers to an
equivalence relation that respects the operation of its underlying semigroup carrier.
Apartially ordered semigroupisasemigroupS withapartialorderďthatrespectsthesemigroup
operation
aďb implies caďcb, acďbc (1.1)
for all elements a,b,cPS. A semigroup S is ordered if the order ď is a total order.
We recall some basic definitions from [14], for the reader’s convenience. As customarily, A˚ denotes
I
thefreemonoidoffinitesequencesgeneratedbyacountablyinfinite totalyorderedsetA :“ta :ℓPIu,
I ℓ
calledalphabet, of letters a ,a ,a ,... An element w PA˚ is calleda word and, unless it is empty, is
1 2 3 I
written uniquely as
w “aq1¨¨¨aqm PA˚, ℓ PI, q PN, (1.2)
ℓ1 ℓm I t t
where a ‰ a for every t. We assume that the empty word, denoted e, belongs to A˚, serving as
ℓt ℓt`1 I
the identity. As customarily, A` stands for the free sub-semigroup obtained from A˚ by excluding the
I I
empty word e. When |I|“n is finite, we write A for the finite alphabet A “ ta ,...,a u. A word is
n I 1 n
ÝÑ
read from left to right, to distinguish this reading direction, when needed, we write w for w.
The free monoid A˚ is endowed with the familiar lexicographic order, denoted ă , induced by
I lx
the order of A . A sub-alphabet A of A is said to be convex, written A Ď A , if J Ď I, i.e.,
I J I I cx I cx
J is a convex subset of I (Definition 1.1). We denote by A the convex sub-alphabet ta ,...,a u of
ri:js i j
A “A .
n r1:ns
The length of a word w PA˚ of the form (1.2) is defined (by the standard summation) as lenpwq :“
I
m q . Then,sincea wordw is afinite sequenceofletters,lenpwqis welldefinedandfinite, withlenpwq
t“1 t
iff w “ e is the empty word e. A word w P A˚ is called k-uniform if each of its letters appears exactly
ř I
k times. We say that w is uniform if w is k-uniform for some k.
AworduPA˚ isafactor ofwPA˚,writtenu|w,ifw“v uv forsomev ,v PA˚. Whenw “v v ,
I I 1 2 1 2 I 1 2
the factors v and v are respectively called the prefix and suffix of w. If v is of length k we say that
1 2 1
v is the k-prefix of w, and denote it by pre pwq. Similarly, if lenpv q“k, we say that v is the k-suffix
i k 2 i
of w, and denote it by suf pwq.
k
Aworduisasubword ofw, writtenuĎ w, ifw canbe writtenasw “v u v u v ¨¨¨u v where
wd 0 1 1 2 2 m m
u and v are words (possibly empty) suchthat u“u u ¨¨¨u , i.e., the u are factors of u. Clearly,any
i i 1 2 m i
factor of w is also a subword, but not conversely.
The following notion of n-power words was introduced in its full generality in [14, §3.1]. However,for
the purpose of the present paper, it suffices to consider a restricted version of this notion, as described
below. Asseenlater,thesen-powerwordsarethecornerstoneofoursystematicconstructionofsemigroup
identities.
Definition 1.2. Let C :“ C be a finite (nonempty) alphabet, and let n P N. An n-power word
m
w :“w is a nonempty word in C` such that:
pC,p,nq m
(a) Each letter a PC may appear in w at most p-times sequentially, i.e., aq ∤w for any q ąp and
ℓ m ℓ
r r a PC ;
ℓ m
(b) Every word uPC` of length n thatrsatisfies rule (a) is a factor of w. r
m
An n-power word is uniform if it is uniform as a word.
r
TROPICAL PLACTIC ALGEBRA 7
An n-power word needs not be unique in C`, and different n-power words may have different length.
m
Often, n-power words w can be concatenated to a new n-power word.
pC,p,nq
Inthepresentpaper,intheviewofTheorem1.8below,wearemostlyinterestedinthecaseofn-power
words over the 2-letter alphabet C “ tx,yu. As we consider powerwords as generic words, we call the
r 2
letters x,y variables.
Example 1.3. Suppose C “tx,yu.
(i) w “yx2y2x is a 2-power uniform word of length 6.
pC,2,2q
(ii) w “xy3xyx3y is a 3-power uniform word of length 10.
pC,3,3q
r
For more details on n-power words see [14, 15].
r
1.3. Co-mirroring and reversal.
Recall that the binary operation of a semigroup is associative.
Definition 1.4. A semigroup homomorphism is a map φ : S ÝÑ S1 that preserves the semigroup
operation, i.e.,
φpa¨bq“φpaq¨φpbq
for all a and b in S. A monoid homomorphism is a semigroup homomorphism for which φpeq “ e1.
It is an endomorphism when S “S1.
A presentation of a monoid M “ pM,¨q (resp. semigroup) is a description of M in terms of a set
of generators A :“ta | ℓPIu and a set of relations ΞĂA˚ˆA˚ on the free monoid A˚ (resp. on the
I ℓ I I I
free semigroup A`) generatedby A . The monoid M is then presented as the quotient A˚{ of the free
I I I Ξ
monoid A˚ by the set of relations Ξ, i.e., by a monoid homomorphism
I
φ:MÝÝÝÝÑA˚{ , ΞĂA˚ˆA˚.
I Ξ I I
When |I|“n is finite, we say that M is finitely presented.
Definition 1.5. The reversal of a word ÝÑw “ a a ¨¨¨a of length m in A˚, ℓ P I, is the word
ÐwÝ“a a ¨¨¨a of the same length m. ℓ1 ℓ2 ℓm I i
ℓm ℓm´1 ℓ1
The reversal ÐwÝ of a word ÝÑw is therefore the rewriting of ÝÑw from right to left, while we formally set
ÐÝ
e :“e. It defines a bijective map
rev:A˚ ÝÝÝÝÑA˚, ÝÑw ÞÝÑÐwÝ,
I I
which is not a monoid homomorphism, since revpuvq ‰ revpuqrevpvq. However, revpuvq “ revpvqrevpuq
for any u,v PA˚.
I
Restricting our ground alphabets to finite alphabets, we introduce the following operation:
Definition 1.6. The co-mirror of a letter a in a finite alphabet A is defined to be the word
ℓ n
1
coM pa q:“a a ¨¨¨a a ¨¨¨a “ a . (1.3)
n ℓ n n´1 n´ℓ`2 n´ℓ 1 t
t“n,
ź
t‰n´ℓ`1
(The co-mirror of the empty word e is formally set to be e.)
The co-mirror of a word w “a ¨¨¨a in A˚ is the word defined as
ℓ1 ℓm n
coM pwq:“ coM pa q¨¨¨ coM pa q, (1.4)
n n ℓ1 n ℓm
written coMpwq when n is clear from the context.
Accordingly, for the concatenation w“uv of two words u,v in A˚ we then have
n
coM pwq:“ coM puqcoM pvq,
n n n
where lenpcoM pwqq “ pn ´1qlenpwq for any w ‰ e. Thus, the co-mirroring of words determines a
n
monoid endomorphism
cmr:A˚ ÝÝÝÝÑA˚, w ÞÝÑ coMpwq, (1.5)
n n
with eÞÝÑe.
8 ZURIZHAKIAN
Remark 1.7. The co-mirroring of a finite alphabet A “ ta ,...,a u introduces a submonoid in the
n 1 n
free monoid A˚, which we denote by pA˚qr1s, whose generators a1 “ coM pa q are again (totaly) ordered
n n ℓ n ℓ
as a1 㨨¨ăa1 . In fact, as can be seen form (1.3), this order is just the lexicographic order ă of A˚,
1 n lx n
and thus is compatible with the initial order of A .
n
Applying the co-mirroring map (1.5) inductively, we have a chain of submonoids, a filtration,
A˚ Ą pA˚qr1s ĄpA˚qr2s Ą pA˚qr3s Ą ¨¨¨ ,
n n n n
together with the surjective homomorphisms
A˚ ÝÝBÝ1։pA˚qr1s ÝÝBÝ2։pA˚qr2s ÝÝBÝ3։pA˚qr3s ÝÝBÝ4։ ¨¨¨ .
n n n n
Each surjection B :pA˚qri´1s Ý։pA˚qris is also injective, preserving the lexicographic order of A˚, and
i n n n
thus it is an isomorphism.
Note that B does not necessarily preserve relations nor presentations over A˚.
i n
1.4. Semigroup identities.
A (nontrivial) semigroup identity is a formal equality of the form
Π:u“v, (1.6)
where u and v are two different (nonempty) words in the free semigroup A`, cf. Form (1.2). For a
I
monoid identity, u and v are allowed to be the empty word as well. Therefore, a semigroup identity Π
determines a single relation ΠPA`ˆA` on the free semigroup A`.
I I I
We say that an identity Π : u “ v is an n-variable identity if u and v involve together exactly n
letters of A . An identity Πis saidto be balanced if the number ofoccurrencesofeachletter a PA is
I i I
the sameinuandinv. Πiscalleduniformly balanced iffurthermorethe wordsuandv arek-uniform
for some k. The length ℓpΠq of Π is defined to be ℓpΠq:“maxtℓpuq,ℓpvqu, clearly ℓpuq“ℓpvq, when Π
is balanced.
A semigroup S :“pS,¨q satisfies the semigroup identity (1.6) if
φpuq“φpvq for every homomorphism φ:A` ÝÑS.
I
Concerning the existence of identities it is suffices to consider 2-variable identities:
Theorem 1.8 ([14, Theorem 3.10]). A semigroup S :“ pS,¨ q that satisfies an n-variable identity Π :
u“v, for ně2, also satisfies a refined 2-variable identity Π:u“v of exponents t1,2u, i.e., each letter
in u and in v appears sequentially at most twice.
p p p
The n-power words w (Definition 1.2) are utilized to construct a class of nontrivial semigroup
pC,p,nq
p p
identities Π . (We use the notation x and y to mark a specific instance of the variables x and y in
pC,p,nq
a given expression, althorugh these instances stand for the same variables x and y, respectively.)
Construction 1.9. Let w be a uniform n-power word over C “ tx,yu such that the words
pC,p,nq
w x w and w y w are both n-power words over C. Define the 2-variable bal-
pC,p,nq pC,p,nq pC,p,nq pC,p,nq
anced identity
r
r r rΠpC,p,nq : r wpC,p,nq x wpC,p,nq “ wpC,p,nq y wpC,p,nq. (1.7)
Then, substitute
xr:“x y rand y :r“y x r (1.8)
to refine (1.7) to the uniformly balanced identity
q q q q
Π : w xy w “ w yx w , (1.9)
pC,p,nq pC,p,nq pC,p,nq pC,p,nq pC,p,nq
where C “tx,yu and wpCq,pq,nq is theqwoqrd obtqaqineqdqfrom wpCq,p,qnq byqsqubsqtitqution (1.8).
NoteqthatqwqpC,p,nq nqeeqds not be an n-power word overr C “ tx,yu, yet it satisfies law (a) of Defini-
tion 1.2. This summarize Construction 1.9, when a semigroup S satisfies the identity (1.9), we shortly
say that it admqitqs the identity (1.9) by taking x“uv and yq“vuqfoqr all elements u,v in S.
Example 1.10 ([14, Example 3.9]). Let C “tx,yu.
TROPICAL PLACTIC ALGEBRA 9
(i) Using the uniform 2-power word w “yx2y2x as in Example 1.3.(i), we receive the identity
pC,2,2q
Π : yx2y2x x yx2y2x “ yx2y2x y yx2y2x . (1.10)
pC,2,2q r
(ii) Taking the uniform 3-power word w “xy3xyx3y in Example 1.3.(ii), we obtain the identity
pC,3,3q
Π : xy3xyx3y x xy3xyx3y “ xy3xyx3y y xy3xyx3y . (1.11)
pC,3,3q r
By substituting x:“xy, y :“yx, these identities (1.10) and (1.11) become uniformly balanced.
1.5. Semirings and semimodules.
qq qq
Equipping a set of elements simultaneously by two binary (monoidial) operations, one obtains the
following structure.
Definition 1.11. A semiring R :“ pR,`,¨q is a set R equipped with two binary operations ` and ¨,
called addition and multiplication, such that:
(i) pR,`q is an Abelian monoid with identity element 0“0 ;
R
(ii) pR,¨q is a monoid with identity element 1“1 ;
R
(iii) Multiplication distributes over addition.
A semiring R is called idempotent if a`a “ a for every a P R, while it is said to be bipotent (or
selective) if a`b P ta,bu for any a,b P R. When the multiplicative monoid pR,¨q is an abelian group,
R is called a semifield.
Clearly, idempotence implies bipotence, but not conversely. A standard general reference for the
structure of semirings is [9].
Remark 1.12. Any ordered monoid pM,¨q gives rise to a semiring, where we define the addition a`b
to be maxta,bu. Indeed, associativity is clear, and distributivity follows from (1.1). In the same way from
an abelian group we obtain a semifield.
Monoid homomorphisms extend naturally to semirings.
Definition 1.13. A homomorphism of semirings is a map ϕ : R ÝÑ R1 that preserves addition and
multiplication. To wit, ϕ satisfies the following properties for all a and b in R:
(i) ϕpa`bq“ϕpaq`ϕpbq;
(ii) ϕpa¨bq“ϕpaq¨ϕpbq;
(iii) ϕp0 q“0 .
R R1
Aunitalsemiringhomomorphism isasemiringhomomorphism thatpreserves themultiplicativeidentity,
i.e., ϕp1 q“1 .
R R1
In the sequel, unless otherwise is specified, our homomorphisms are assumed to be unital.
In analogy to the case of rings, one defines module over semiring in a straightforwardway.
Definition1.14. AnR-moduleV over asemiringRis an additive monoid pV,`qtogether with ascalar
multiplication RˆV ÑV satisfying the following properties for all r PR and v,w PV:
i
(i) rpv`wq“rv`rw;
(ii) pr `r qv “r v`r v;
1 2 1 2
(iii) pr r qv “r pr vq;
1 2 1 2
(iv) 1 v “v;
R
(v) 0 v “0 “r0 .
R V V
Modules over semirings are also called semimodules.
2. Tropical plactic algebra
On of the central ground objects of the current paper is the following well-known monoid [30]:
10 ZURIZHAKIAN
Definition 2.1. The plactic monoid is the monoid P :“ PLCpA q, generated by an ordered set of
I I
elements A :“ta | ℓPIu, subject to the equivalence relation ” (known as the elementary Knuth
I ℓ knu
relations or the plactic relations) defined by
KN1. acb“cab if aďbăc,
(KNT)
KN2. bac“bca if aăbďc,
for all triplets a,b,c P A . Namely, P :“ A˚{ with the identity element e. When |I| “ n is finite,
n I I ”knu
P is said to be of rank n, or finitely generated, and is denoted by P . 1
I n
WewritePLC forPLCpA qwhenthealphabetA isarbitrary. Henceforth,wealwaysassumethateăa
I I I ℓ
for all a PP . The relation ” is a congruence on the free monoid A˚, e.g. see A˚ [29], which is also
ℓ I knu I I
denoted by ” to indicate its relevance to PLC .
plc I
We aim for an algebra whose multiplicative structure comprises that of the plactic monoid, and in
which the Knuth relations are inferred from the algebraic operations. Towards this goal, we begin with
anaxillarysemigroupstructure, to be employedfor a constructionofsucha desirable algebra– a special
semiring. (In what follows we let I ĂN be a nonempty subset.)
Definition 2.2. A (partial) forward semigroup F “ xf | iPIy is a pointed semigroup with an
I i
absorbing element o, generated by the (partial) ordered set of elements 2 tf | iPIu, subject to the axiom
i
FS: f f “o whenever f ąf .
j i j i
When I is finite with |I|“n, we write F for F .
n I
Note that the semigroup F is not assumed to be a (partial) ordered semigroup, i.e., having an order
I
preserving operation f ě f ñ f f ě f f (Definition 1.4), nor a cancellative semigroup, i.e., f f “
i j i k j k k i
f f ; f “ f . Usually, F is non-commutative, since otherwise we would always have f f “ f f “ o,
k j i j I i j j i
implying that F consists of tf | iPIuYtou.
I i
Remark 2.3. The elements of a forward semigroup F are realized as nondecreasing sequences over
I
tf | i P Iu. Then it easy to verify that F naively admits the Knuth relations (KNT), since in F each
i I I
term of (KNT) equals o.
Clearly, the forward semigroup F is a very degenerated version of the plactic monoid, but useful for
I
the construction of semirings, as seen below in Construction 2.6.
2.1. Tropical plactic algebra.
We open this section by introducing a new semiring structure – a key object in this paper – that
encapsulates the plactic monoid (cf. Theorem 2.8 below).
Definition2.4. Atropical plactic algebraplc isa(noncommutative)idempotentsemiringpplc ,`,¨q
I I
with multiplicative identity element e and zero o, generated by an ordered set of elements 2 ta | i P Iu
i
subject to the axioms (for every aďbďc in ta | iPIu):
i
PA1: a“e`a;
PA2: ba“a`b when bąa;
PA3: apb`cq“ab`c;
PA4: pa`bqc“a`bc.
When |I|“n is finite we write plc for plc , and say that plc is finitely generated and has rank n. (For
n I n
indexing matter, e is also denoted by a .)
0
We abbreviate our terminology and also write troplactic algebra for tropical plactic algebra. Arbi-
trary elements of plc are denoted by Gothic letters u,v,a, while a,b,c, are devoted for its generators.
I
When it is clear from the context, we suppress the notation of the indexing set I and write plc for plc ,
I
for short.
1In the literature, PLCI is often assumed to finitely generated, however for large parts of our study are more general
andfiniteness isnotrequiredthere.
2Forsimplicity,weassumeacountablesetofgenerators,wherethegeneralizationtoanarbitraryorderedsetofgenerators
is obvious. Moreover, one can also generalize this structure by considering a partiallyordered set of generators, but such
theoryismoreinvolved.