From plactic monoids to hypoplactic monoids

The plactic monoids can be obtained from the tensor product of crystals. Similarly, the hypoplactic monoids can be obtained from the quasi-tensor product of quasi-crystals. In this paper, we present a unified approach to these constructions by expressing them in the context of quasi-crystals. We provide a sufficient condition to obtain a quasi-crystal monoid for the quasi-tensor product from a quasi-crystal monoid for the tensor product. We also establish a sufficient condition for a hypoplactic monoid to be a quotient of the plactic monoid associated to the same seminormal quasi-crystal.


Introduction
The plactic monoid, formally introduced by Lascoux and Schützenberger [LS81], is an algebraic object of great interest, with connections to several fields such as representation theory, combinatorics [Ful97], symmetric functions, and Schubert polynomials [LS85,LS89].It was also used to give a first rigorous proof of the Littlewood-Richardson rule [LR34].It originally emerged from Young tableaux and the Schensted insertion algorithm [Sch61] with a presentation given by the Knuth relations [Knu70].
Kashiwara [Kas90,Kas91,Kas94], following the work by Date, Jimbo and Miwa [DJM90], introduced crystal bases for modules of quantized universal enveloping algebras (also known as quantum groups), discovered independently by Drinfel'd [Dri85] and Jimbo [Jim85].Crystal bases can be described by weighted labelled graphs that are called crystal graphs.Kashiwara showed that the plactic monoid arises from the crystal basis associated with the vector representation of the quantized universal enveloping general linear Lie algebra by identifying elements in the same position of isomorphic connected components of the associated crystal graph.This result allowed a deeper study of the plactic monoid and its generalization, because the underlying construction still results in a monoid for crystal bases associated with other quantized universal enveloping algebras, as it only relies on the definition of seminormal crystals and their tensor product.Thus, based on the work by Kashiwara and Nakashima [KN94], Lecouvey [Lec02,Lec03] presented comprehensive descriptions of the plactic monoids for the Cartan types B n , C n , and D n , which later appeared in a survey [Lec07].In recent works, Cain, Gray and Malheiro [CGM15a,CGM19] presented rewriting systems and biautomatic structures for these monoids.
The hypoplactic monoid, introduced by Krob and Thibon [KT97] and studied by Novelli [Nov00], emerged from a noncommutative realization of quasi-symmetric functions analogous to the realization of symmetric functions by the plactic monoid presented by Lascoux and Schützenberger [LS81].It was originally obtained from quasi-ribbon tableaux and an insertion algorithm with a presentation consisting of the Knuth relations and the quartic relations.A comparative study with other monoids was done by Cain, Gray and Malheiro in [CGM15b], where a rewriting system and a biautomatic structure for the hypoplactic monoid is presented.Recently, Cain, Malheiro and Ribeiro [CMR22] provide a complete description of the identities satisfied by the hypoplactic monoid.
To obtain a construction of the hypoplactic monoid analogous to the construction of the plactic monoid from crystals, a first notion of quasi-crystal graph was introduced by Krob and Thibon [KT99].To overcome some limitations of this notion, Cain and Malheiro [CM17] described a new notion of quasi-crystal graph, which is equivalent to another notion considered recently by Maas-Gariépy [MG23], from which the hypoplactic monoid arises by identifying words in the same position of isomorphic connected components.These notions of quasi-crystal graphs are based on the crystal graph for Cartan type A n , and the construction does not result in a monoid if this crystal graph is replaced by the crystal graph for another Cartan type.
Cain, Malheiro, and the present author [CGM23] introduced a general notion of quasi-crystals associated to a root system.This notion is further studied in [CMRR23].It allows the construction of the hypoplactic monoid by identifying words in the same position of isomorphic connected components of a quasi-crystal associated to type A n .Moreover, it allows the generalization of the hypoplactic monoid, because this construction still results in a monoid for any other seminormal quasi-crystal, as it only relies on the definition of quasi-tensor product of seminormal quasi-crystals, also introduced in [CGM23].
For this notion, crystals are quasi-crystals; in particular, the class of seminormal crystals is contained on the class of seminormal quasi-crystals.Also, the definition of tensor product of seminormal crystals extends in a natural way to seminormal quasi-crystals.Thus, quasi-crystals form a framework where the constructions of plactic monoids and hypoplactic monoids can be expressed.
The aim of this paper is to formalize and develop this unified approach where plactic and hypoplactic monoids are associated with the same combinatorial objects: quasi-crystals.This contrasts with previous approaches in the literature, which obtain independently plactic monoids from crystals and hypoplactic monoids from (some notion of) quasi-crystals.As shown by this unified construction, plactic and hypoplactic monoids emerge from tensor and quasi-tensor products of quasi-crystals, respectively.As the underlying combinatorial objects are the same, it allows a comparative study between plactic and hypoplactic monoids, where some well-known constructions for the classical case are shown to be consequence of properties of the underlying quasi-crystals.This unified construction also allows a deeper understanding of the results obtained in [CGM23] for the relation between the plactic and hypoplactic monoids of type C n .
This paper is structured as follows.Section 2 introduces the necessary background on root systems and quasi-crystals.Section 3 presents a construction of the plactic monoid based on the tensor product of quasi-crystals, and Section 4 presents an analogous construction of the hypoplactic monoid based on the quasi-tensor product of quasi-crystals.From these sections, plactic and hypoplactic monoids are obtained as a quotient of the free quasicrystal monoid for the respective product of quasi-crystals.Section 5 shows a construction of quasi-crystal monoids for the quasi-tensor product from quasi-crystal monoids for the tensor product, whenever the monoids are equidivisible.When applied to the free quasi-crystal monoids, it results in the constructions described in [CM17,CGM23].In Section 6, it is given a sufficient condition for the hypoplactic monoid to be a quotient of the plactic monoid associated to the same quasi-crystal.For type A n , this result is an alternative prove of the well-known fact that the classical hypoplactic monoid is a quotient of the classical plactic monoid, and for type C n , it is a step towards understanding the relation between plactic and hypoplactic monoids for this type.

Preliminaries
In this section, we present the necessary background on crystals and quasicrystals.We will introduce crystals as a subclass of quasi-crystals, so we will not need to present a complete introduction to crystals.We refer to [Kas95] for an introduction to crystals as they originally emerged in connection to quantized universal enveloping algebras (also called quantum groups) or [HK02] for a comprehensive background on this approch, and to [BS17] for a study of crystals detached from their origin, For a detailed study of quasicrystals, see [CGM23].
2.1.Root systems.We first give the essential background on root systems, as these algebraic structures will be used to define crystals and quasicrystals.Root systems are commonly found in representation theory, in particular, they arise on the study of Lie groups and Lie algebras, but we will detach them from this context, as we will only introduce what we need for our purpose.For further context see for example [FH91,EW06,Bum13].
Let V be a Euclidean space, that is, a real vector space with an inner product • , • .For α ∈ V other than 0, denote by r α the reflection in the hyperplane orthogonal to α, which is given by for each v ∈ V .Note that r α is bijective, as r α r α (v) = v, for all v ∈ V .Also, r α preserves the inner product, as r α (u), r α (v) = u, v for any u, v ∈ V .
Definition 2.1.A root system in V is a subset Φ of V satisfying the following conditions: (1) Φ is nonempty, finite, and 0 The elements of Φ are called roots, and the elements α ∨ , with α ∈ Φ, are called coroots.
Together with a root system, we always fix an index set I and simple roots (α i ) i∈I , that is, a collection of roots satisfying the following conditions: • every root β ∈ Φ can be expressed as β = i∈I k i α i , where all k i are either nonnegative or nonpositive integers.Finally, together with a root system we also consider the following structure.
The elements of Λ are called weights.
In the subsequent sections, we always consider a root system Φ with weight lattice Λ and index set I for the simple roots (α i ) i∈I .The only non-arbitrary root systems that will be considered are the root systems associated to Cartan types A n and C n .
Let n ≥ 2. Consider V to be the real vector space R n with the usual inner product, and denote by e i ∈ R n the n-tuple with 1 in the i-th position, and 0 elsewhere, i = 1, 2, . . ., n.The root system associated to Cartan type A n based on the general linear Lie algebra gl n consists of Φ = {e i − e j | i = j}, the index set for the simple roots is I = {1, 2, . . ., n − 1}, the simple roots are α i = e i − e i+1 , i = 1, 2, . . ., n − 1, and the weight lattice is Λ = Z n .For this type, we follow the notation used in [Lec02,CGM19], although it is also commonly denoted in the literature by A n−1 .
For more examples of root systems see [ (1) φi (x) = εi (x) + wt(x), The set Q is called the underlying set of Q, and the maps wt, ëi , fi , εi and φi (i ∈ I) form the quasi-crystal structure of Q.Also, the map wt is called the weight map, where wt(x) is said to be the weight of x ∈ Q, and the maps ëi and fi (i ∈ I) are called the raising and lowering quasi-Kashiwara operators, respectively.
In this definition, ⊥ is an auxilary symbol.For x ∈ Q, by ëi (x) = ⊥ (or fi (x) = ⊥) we mean that ëi (resp., fi ) is undefined on x.On the other hand, we say that ëi (or fi ) is defined on x whenever ëi (x) ∈ Q (resp., fi (x) ∈ Q).Thus, alternatively one can consider the quasi-Kashiwara operators ëi and fi (i ∈ I) to be partial maps from Q to Q.When this point of view is more suitable to describe quasi-Kashiwara operators, we will make use of it.
As in this paper we will only consider quasi-crystals that are seminormal, i.e., quasi-crystals satisfying condition (6) above, we included it directly in the definition.In the sequel, when we only write "quasi-crystal", we always mean "seminormal quasi-crystal".
As we now describe, a seminormal quasi-crystal can be completely encoded in a graph.See [CGM23, § 4] for further background.Definition 2.4.Let Φ be a root system with weight lattice Λ and index set I for the simple roots (α i ) i∈I .The quasi-crystal graph Γ Q of a quasi-crystal Q of type Φ is a Λ-weighted I-labelled directed graph with vertex set Q and an edge x i −−→ y from x ∈ Q to y ∈ Q labelled by i ∈ I whenever fi (x) = y, and a loop on x ∈ Q labelled by i ∈ I whenever εi (x) = +∞.For x ∈ Q, let Γ Q (x) denote the connected component of Γ Q containing the vertex x.
A crystal graph is a quasi-crystal graph that does not have loops.Thus, we have that Γ Q is a crystal graph if and only if Q is a crystal.
Example 2.5.Consider the root system of type A n .The standard crystal A n of type A n is given as follows.The underlying set is the ordered set For i = 1, 2, . . ., n − 1, the Kashiwara operators ẽi and fi are only defined on i + 1 and i, respectively, where ẽi (i + 1) = i and fi (i) = i + 1.Finally, εi (x) = δ x,i+1 and φi (x) = δ x,i , where δ k,l = 1 if k = l, and δ k,l = 0 whenever k = l.
The crystal graph Γ An is where the weight map is defined as above.
Example 2.6.Consider the root system of type C n .The standard crystal C n of type C n is defined as follows.The underlying set is the ordered set the weight of x is wt(x) = e x , and the weight of x is wt(x) = −e x .For i = 1, 2, . . ., n − 1, the Kashiwara operators ẽi and fi are only defined on the following cases: ẽi (i + 1) = i, ẽi (i) = i + 1, fi (i) = i + 1, and fi (i + 1) = i.The Kashiwara operators ẽn and fn are only defined on n and n, respectively, where ẽn (n) = n and fn (n , and φn (y where the weight map is defined as above.
To conclude, we introduce the notion of homomorphism between seminormal quasi-crystals.
Definition 2.7.Let Q and Q ′ be seminormal quasi-crystals of the same type.A quasi-crystal homomorphism {⊥} that satisfies the following conditions: (1) If ψ is also bijective, then it is called a quasi-crystal isomorphism.We say that Q and Q ′ are isomorphic if there exists a quasi-crystal isomorphism between Q and Q ′ .
In terms of quasi-crystal graphs, we have that two quasi-crystals Q and Q ′ of the same type are isomorphic if and only if there exists a graph isomorphism between Γ Q and Γ Q ′ that preserves vertex weights and edge labels.

Plactic monoids over quasi-crystals
In this section, we present a construction of the plactic monoid associated to a seminormal quasi-crystal.It is based on the well-known construction of the plactic monoid associated to a seminormal crystal, which has its roots in the work by Kashiwara and Nakashima [KN94] and is commonly found in the literature.See for example [Lec07,CGM19].
We first define the tensor product of seminormal quasi-crystals.We then introduce quasi-crystal monoids for this tensor product and define a free quasi-crystal monoid associated to a seminormal quasi-crystal.Finally, we show how the plactic monoid can be defined as a quotient of the free quasicrystal monoid.
Example 3.3.The crystal graph Γ C 2 ⊗C 2 of the tensor product C 2 ⊗ C 2 is the following.
We chose to base our definition of tensor product of seminormal quasicrystals on the original notion of tensor product of crystals introduced by Kashiwara [Kas90, Kas91,Kas94].There is another convention for the tensor product of crystals which is opposite to the original one (see for example [BS17, § 2.3]).The impact of this choice in the subsequent sections is that the monoids considered are anti-isomorphic to the ones that would be obtained if we chose to adopt the other convention.
3.2.Quasi-crystal monoids for the tensor product.We first introduce the fundamental concept relating quasi-crystals and monoids with respect to the tensor product.Definition 3.4.Let Φ be a root system with weight lattice Λ and index set I for the simple roots (α i ) i∈I .A ⊗-quasi-crystal monoid M of type Φ consists of a set M together with maps wt : M → Λ, ëi , fi : M → M ⊔ {⊥}, εi , φi : M → Z ∪ {+∞} (i ∈ I) and a binary operation • : M × M → M satisfying the following conditions: (1) M together with wt, ëi , fi , εi and φi (i ∈ I) forms a seminormal quasi-crystal of type Φ; (2) M together with • forms a monoid; (3) the map M ⊗ M → M , given by x ⊗ y → x • y for x, y ∈ M , induces a quasi-crystal homomorphism from M ⊗ M to M.
In a ⊗-quasi-crystal monoid M the interaction between the quasi-crystal structure and the binary operation satisfies rules similar to those satisfied by the quasi-crystal structure of a tensor product.That is, φi (xy) = max φi (x) + wt(y), α ∨ i , φi (y) , for x, y ∈ M and i ∈ I.
When M together with wt, ëi , fi , εi and φi (i ∈ I) forms a seminormal crystal, we say that M is a ⊗-crystal monoid.
Given a seminormal quasi-crystal Q, as the tensor product ⊗ is an associative operation, denote by Q ⊗k the tensor product of k copies of Q and by Q ⊗k its underlying set. the set inherits a ⊗-quasi-crystal monoid structure, which gives rise to the following definition.
Notice that we explicitly gave the values of the quasi-crystal structure maps of Q ⊗ on the empty word ǫ, on letters the values follow from the quasi-crystal structure maps of Q, and on a word of the form uv they depend only on their values on u and v. Thus, the definition of the quasi-crystal structure above is not circular.Moreover, we can obtain the values of the quasi-crystal structure maps on a nonempty word based only on their values on its letters.For the weight map, we have that for any x 1 , . . ., x m ∈ Q.For ëi , fi , εi , and φi (i ∈ I), we present a method called signature rule.
Let B 0 denote the bicyclic monoid with a zero element added.We have the following presentation where we omitted the relations (x0, 0) and (0x, 0), for x ∈ {0, −, +}, for the sake of simplicity.
Let Q be a seminormal quasi-crystal.For each i ∈ I, define a map sgn ⊗ i : i is called the i-signature map for the tensor product ⊗.
The i-signature map sgn ⊗ i is a monoid homomorphism.Thus, given a word w = x 1 . . .x m with x 1 , . . ., x m ∈ Q, we have that We compute ẽi , fi , εi , and φi on w = 313122414, for i ∈ {1, 2, 3}, using the signature rule.To keep track to which element originates each − and + we write a subscript with the position of the letter, this is just an auxiliary notation and the binary operation of B 0 should be applied ignoring the subscripts.

Plactic monoids.
Given a seminormal quasi-crystal Q and an element x ∈ Q, we denote by Q(x) the seminormal quasi-crystal whose quasi-crystal graph corresponds to the connected component Γ Q (x) of Γ Q containing x.We call Q(x) the connected component of Q containing x, and denote its underlying set by Q(x).Thus, the quasi-crystal structure of Q(x) corresponds to the restriction of the quasi-crystal structure of Q to Q(x).Definition 3.7.Let Q be a seminormal quasi-crystal.The plactic congruence on Q ⊗ is a relation ≈ on Q * given as follows.For u, v ∈ Q * , u ≈ v if and only if there exists a quasi-crystal isomorphism ψ : Due to the following result, we can define the plactic monoid over a quasicrystal.
Proposition 3.8.Let Q be a seminormal quasi-crystal.Then, the plactic congruence ≈ on Q ⊗ is a monoid congruence on the free monoid Q * .Definition 3.9.Let Q be a seminormal quasi-crystal, and let ≈ be the plactic congruence on Q ⊗ .The quotient monoid Q * /≈ is called the plactic monoid associated to Q and is denoted by plac(Q).
Observe that the quasi-crystal structure of Q ⊗ induces a quasi-crystal structure on plac(Q), since if u ≈ v, then wt(u) = wt(v), εi (u) = εi (v), φi (u) = φi (v), ëi (u) ≈ ëi (v) whenever ëi is defined on u, and fi (u) ≈ fi (v) whenever fi is defined on v (i ∈ I).Although plac(Q) is a ⊗-quasi-crystal monoid, we simply call it the plactic monoid, because we are mainly interested in studying it as a monoid.However, we will be constantly considering its quasi-crystal structure, as it plays a fundamental role in the construction of plac(Q), and consequently, in its properties.
For type A n , we have that the plactic monoid plac(A n ) is anti-isomorphic to the classical plactic monoid introduced by Lascoux and Schützenberger (1), described in Example 2.6.For instance, we get that 122 ≈ 1, 131 ≈ 3, and 22 1 ≈ 1.

Hypoplactic monoids over quasi-crystals
In this section, we present a construction of the hypoplactic monoid associated to a seminormal quasi-crystal.It is analogous to the construction described in Section 3.For a detailed study of this construction and for the proofs of all results in this section, see [CGM23].
We first define the quasi-tensor product of seminormal quasi-crystals.We then introduce quasi-crystal monoids and free quasi-crystal monoids for the quasi-tensor product.Finally, we show how the plactic monoid can be defined as a quotient of the free quasi-crystal monoid.
4.1.Quasi-tensor product of quasi-crystals.We present the notion of quasi-tensor product of seminormal quasi-crystals.A detailed study of this notion can be found in [CGM23, § 5].Definition 4.1.Consider a root system Φ with weight lattice Λ and index set I for the simple roots (α i ) i∈I .Let Q and Q ′ be seminormal quasi-crystals of type Φ.The inverse-free quasi-tensor product of Q and Q ′ , or simply the quasi-tensor product of Q and Q ′ , is the seminormal quasi-crystal Q ⊗ Q ′ given as follows.The underlying set is Q ⊗Q ′ which consists of the Cartesian product Q × Q ′ whose ordered pairs are denoted by x ⊗ x ′ with x ∈ Q and x ′ ∈ Q ′ .The quasi-crystal structure is defined by wt(x ⊗ x ′ ) = wt(x) + wt(x ′ ), and (1) if φi (x) > 0 and εi (x ′ ) > 0, then ëi (x ⊗ x ′ ) = fi (x ⊗ x ′ ) = ⊥ and εi (x ⊗ x ′ ) = φi (x ⊗ x ′ ) = +∞; (2) otherwise, The quasi-tensor product of seminormal quasi-crystals is associative, that is, given seminormal quasi-crystals Q 1 , Q 2 , and Q 3 of the same type, the quasi-crystals ( Observe that the quasi-tensor product of seminormal crystals may not be a seminormal crystal.In fact, given a crystal Q, if x ∈ Q and i ∈ I are such that fi is defined on x, then φi (x) > 0 and εi fi (x) = εi (x) + 1 > 0, which implies that εi x ⊗ fi (x) = +∞, and so Q ⊗ Q is not a crystal.
Comparing the notions of tensor product (Definition 3.1) and quasi-tensor product, we have the following observation.
Remark 4.2.Let Q and Q ′ be seminormal quasi-crystals of the same type, and let x ∈ Q and Example 4.3.The quasi-crystal graph Γ A 3 ⊗A 3 of the quasi-tensor product A 3 ⊗ A 3 is the following.
Example 4.4.The quasi-crystal graph Γ C 2 ⊗C 2 of the quasi-tensor product C 2 ⊗ C 2 is the following.
4.2.Quasi-crystal monoids for the quasi-tensor product.We first introduce the fundamental concept relating quasi-crystals and monoids with respect to the quasi-tensor product.
and φi (xy) = max φi (x) + wt(y), α ∨ i , φi (y) , Given a seminormal quasi-crystal Q, as the quasi-tensor product ⊗ is an associative operation, denote by Q ⊗k the quasi-tensor product of k copies of Q and by Q ⊗k its underlying set. the set inherits a ⊗-quasi-crystal monoid structure, which gives rise to the following definition.
In a free ⊗-quasi-crystal monoid Q ⊗, we can obtain the values of the quasi-crystal structure maps on a nonempty word based only on their values on its letters, which are given by the quasi-crystal structure of Q.For the weight map, we have that wt(x 1 . . .x m ) = wt(x 1 ) + • • • + wt(x m ).
Let Q be a seminormal quasi-crystal.For each i ∈ I, define a map sgn ⊗ i : i is called the i-signature map for the quasi-tensor product ⊗.
The i-signature map sgn ⊗ i is a monoid homomorphism.Thus, given a word w = x 1 . . .x m with x 1 , . . ., x m ∈ Q, we have that We compute ëi , fi , εi , and φi on w = 211213, for i ∈ {1, 2, 3}, using the signature rule for the quasi-tensor product ⊗.To keep track to which element originates each − and + we write a subscript with the position of the letter, this is just an auxiliary notation and the binary operation of Z 0 should be applied ignoring the subscripts.
4.3.Hypoplactic monoids.We now show how the hypoplactic monoid arises by identifying words in the same position of isomorphic connected components of a free ⊗-quasi-crystal monoid.
Due to the following result, we can define the hypoplactic monoid over any seminormal quasi-crystal.Proposition 4.9 ([Gui22, Theorem 8.23]).Let Q be a seminormal quasicrystal.Then, the hypoplactic congruence ∼ on Q ⊗ is a monoid congruence on the free monoid Q * .Definition 4.10.Let Q be a seminormal quasi-crystal, and let ∼ be the hypoplactic congruence on Q ⊗.The quotient monoid Q * / ∼ is called the hypoplactic monoid associated to Q and is denoted by hypo(Q).
As we pointed out for the plactic monoid, we have that the quasi-crystal structure of Q ⊗ induces a quasi-crystal structure on hypo(Q).Although hypo(Q) is a ⊗-quasi-crystal monoid, we simply call it the plactic monoid, because we are mainly interested in studying it as a monoid.However, we will be constantly considering its quasi-crystal structure, as it plays a fundamental role in the construction of hypo(Q), and consequently, in its properties.
For type A n , we have that the hypoplactic monoid hypo(A n ) is antiisomorphic to the classical hypoplactic monoid introduced by Krob  .We get that 11 ∼ 1111 ∼ 1111.

From ⊗-quasi-crystal monoids to ⊗-quasi-crystal monoids
In this section, we investigate wether a ⊗-quasi-crystal monoid gives rise to a ⊗-quasi-crystal monoid.For this purpose, we consider the following monoid property.Definition 5.1.A monoid M is said to be equidivisible if for any elements x 1 , x 2 , y 1 , y 2 ∈ M satisfying x 1 y 1 = x 2 y 2 , there exists z ∈ M such that x 2 = x 1 z and y 1 = zy 2 , or such that x 1 = x 2 z and y 2 = zy 1 .
The result for ëi follows analogously.
Lemma 5.3.Let M be a ⊗-quasi-crystal monoid which is equidivisible, and let x, y, x ′ , y ′ ∈ M be such that xy = x ′ y ′ .For i ∈ I, if φi (x) > 0 and εi (y) > 0, then one of the following conditions holds: (1) Proof.Take i ∈ I such that φi (x) > 0 and εi (y) > 0. As M is equidivisible, there exists z ∈ M such that one of the following cases hold.
It remains to show that the map ψ : M ⊗M → M , given by ψ(x ⊗ y) = xy for each x, y ∈ M , is a quasi-crystal homomorphism from M ⊗ ⊗M ⊗ to M ⊗.

and φ
⊗ i on x, y, and xy coincide with the values of ë⊗ i , f ⊗ i , ε⊗ i , and φ⊗ i on x, y, and xy, respectively; and since M ⊗ is a ⊗-quasi-crystal monoid, we obtain from Remark 4.2 that ψ satisfies the axioms of Definition 2.7.Otherwise, we get one of the following cases.
• Case 1: x = x 1 x 2 for some x 1 , x 2 ∈ M such that φ⊗ i (x 1 ) > 0 and ε⊗ 1 (x 2 ) > 0.Then, ε • Case 2: y = y 1 y 2 for some y 1 , y 2 ∈ M such that φ⊗ i (y 1 ) > 0 and ε⊗ i (y 2 ) > 0.Then, ε • Case 3: xy = z 1 z 2 for some z 1 , z 2 ∈ M such that φ⊗ i (z 1 ) > 0 and ε⊗ i (z 2 ) > 0. By Lemma 5.3, we have case 1, case 2, or φ⊗ i (x) > 0 and ε⊗ i (y) > 0.Then, ε Let Q be a seminormal quasi-crystal.From the previous result, we can easily obtain the quasi-crystal graph Γ Q ⊗ of the free ⊗-quasi-crystal monoid Start with Γ Q ⊗ ; then, for each word w ∈ Q * and each i ∈ I such that w = uv, for some u, v ∈ Q * where u is the start of an i-labelled edge and v is the end of an i-labelled edge, remove any i-labelled edge starting or ending on w, and then add an i-labelled loop on w; thus, the resulting graph is Γ Q ⊗ .
Let Q be a seminormal quasi-crystal.From the previous remark, we have that the quasi-crystal graph Γ Q ⊗ of the free ⊗-quasi-crystal monoid Q ⊗ can be constructed from the quasi-crystal graph Γ Q ⊗ of the free ⊗-quasi-crystal monoid Q ⊗ as follows.Start with Γ Q ⊗ ; then, for each word w ∈ Q * and each i ∈ I such that w = u 1 xu 2 yu 3 , for some u 1 , u 2 , u 3 ∈ Q * and x, y ∈ Q where x is the start of an i-labelled edge and y is the end of an i-labelled edge, remove any i-labelled edge starting or ending on w, and then add an i-labelled loop on w; thus, the resulting graph is Γ

Hypoplactic monoids as quotients of plactic monoids
From the presentation for the classical hypoplactic monoid given by Krob and Thibon [KT97], which contains the Knuth relations [Knu70], it is immediate that the hypoplactic monoid hypo(A n ) of type A n is a quotient of the plactic monoid plac(A n ) of type A n .On the other hand, as shown in [CGM23], the hypoplactic monoid hypo(C n ) of type C n contains submonoids isomorphic to hypo(A n−1 ) and hypo(C n−1 ), but it is not a quotient of the plactic monoid plac(C n ) of type C n .Understanding the properties that originate these results is an active research problem.
The converse implication follows analogously.
Proposition 6.2.Let Q be a seminormal quasi-crystal such that wt(x) x ∈ Q is a linearly independent set.Then, ≈ ⊆ ∼.Therefore, hypo(Q) is a monoid quotient of plac(Q).
For the standard crystal A n of type A n , the weights wt(x) = e x , x ∈ A n , are linearly independent.Thus, from the previous proposition, we get the well-known result that the classical hypoplactic monoid is a quotient of the classical plactic monoid.

3. 1 .
Tensor product of quasi-crystals.We present the notion of tensor product of seminormal quasi-crystals following the original notion of tensor product of crystals introduced by Kashiwara [Kas90, Kas91, Kas94].Definition 3.1.Consider a root system Φ with weight lattice Λ and index set I for the simple roots ( Consider the standard crystal A 3 of type A 3 .The connected component Γ A ⊗ 3 For instance, we have that 112 ≈ 121, 212 ≈ 122, 223 ≈ 232, and 323 ≈ 233.Example 3.11.Consider the standard crystal C 3 of type C 3 .The connected component Γ isomorphic to the connected component Γ C ⊗ 3

−−→ 121 and 12 1 1 −
with the one given by Cain and Malheiro in [CM17].Example 5.6.Consider the standard crystal A 3 of type A 3 .From the connected component Γ A ⊗ 3 (112), described in Example 3.10, we remove the 1-labelled edge 112 1 −−→ 212; add 1-labelled loops on 112, 212, and 312; remove the 2-labelled edge 223 2 −−→ 323; and adde 2-labelled loops on 213, 223, and 323.The resulting quasi-crystal graph is to the connected components Γ A ⊗ 3 (112) and Γ A ⊗ 3 (212).Example 5.7.Consider the standard crystal C 3 of type C 3 .From the connected component Γ C ⊗ 3 122 , described in Example 3.11, we remove the 1-labelled edges 122 1 −→ 22 1; add 1-labelled loops on each vertex; and adde 2-labelled loops on 122 and 22 1.The resulting quasi- Let Φ be a root system with weight lattice Λ and index set I for the simple roots (α i ) i∈I .A seminormal quasi-crystal Q of type Φ consists of a set Q together with maps wt : Q → Λ, ëi , fi : Q → Q ⊔ {⊥} and εi , φi : Q → Z ∪ {+∞}, for each i ∈ I, satisfying the following conditions: [KT97]ibon[KT97].Consider the standard crystal A 3 of type A 3 .The connected component Γ A ⊗ Consider the standard crystal C n of type C n .The connected components Γ C ⊗ (x) does not admit a decomposition of the form x 1 x 2 with x 1 , x 2 ∈ M such that φ⊗ i (x 1 ) > 0 and ε⊗ i (x 2 ) > 0, because x = f ⊗ i ∈ I.If u ′ and v ′ do not admit a decomposition of the form w 1 xw 2 yw 3 with w 1 , w 2 , w 3 ∈ Q * and x, y ∈ Q such that φ ′ and v ′ coincide with the values of ë⊗ i , f ⊗ i , ε⊗ i , and φ⊗ i on u ′ and v ′ , respectively, which implies that ψ ⊗ ë ⊗i on u