Structure of relatively free trioids
Loday and Ronco introduced the notions of a trioid and a trialgebra, and constructed the free trioid of rank 1 and the free trialgebra. This paper is a survey of recent developments in the study of free objects in the varieties of trioids and trialgebras. We present the constructions of the free tri...
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Zhuchok, A.V. 2023-03-11T13:31:29Z 2023-03-11T13:31:29Z 2021 Structure of relatively free trioids / A.V. Zhuchok // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 1. — С. 152–166. — Бібліогр.: 35 назв. — англ. 1726-3255 DOI:10.12958/adm1732 2020 MSC: 08B20, 20M10, 20M50, 17A30, 17D99. https://nasplib.isofts.kiev.ua/handle/123456789/188682 Loday and Ronco introduced the notions of a trioid and a trialgebra, and constructed the free trioid of rank 1 and the free trialgebra. This paper is a survey of recent developments in the study of free objects in the varieties of trioids and trialgebras. We present the constructions of the free trialgebra and the free trioid, the free commutative trioid, the free n-nilpotent trioid, the free left (right) n-trinilpotent trioid, and the free rectangular trioid. Some of these results can be applied to constructing relatively free trialgebras. Dedicated to the 60th anniversary of the Department of Algebra and Mathematical Logic of Taras Shevchenko National University of Kyiv The author is supported by the National Research Foundation of Ukraine (grant no. 2020.02/0066). en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Structure of relatively free trioids Article published earlier |
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Loday and Ronco introduced the notions of a trioid and a trialgebra, and constructed the free trioid of rank 1 and the free trialgebra. This paper is a survey of recent developments in the study of free objects in the varieties of trioids and trialgebras. We present the constructions of the free trialgebra and the free trioid, the free commutative trioid, the free n-nilpotent trioid, the free left (right) n-trinilpotent trioid, and the free rectangular trioid. Some of these results can be applied to constructing relatively free trialgebras.
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Structure of relatively free trioids / A.V. Zhuchok // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 1. — С. 152–166. — Бібліогр.: 35 назв. — англ. |
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“adm-n1” — 2021/4/10 — 20:38 — page 152 — #156
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 31 (2021). Number 1, pp. 152–166
DOI:10.12958/adm1732
Structure of relatively free trioids
A. V. Zhuchok∗
Dedicated to the 60th anniversary of the Department of Algebra and
Mathematical Logic of Taras Shevchenko National University of Kyiv
Abstract. Loday and Ronco introduced the notions of
a trioid and a trialgebra, and constructed the free trioid of rank 1
and the free trialgebra. This paper is a survey of recent developments
in the study of free objects in the varieties of trioids and trialgebras.
We present the constructions of the free trialgebra and the free
trioid, the free commutative trioid, the free n-nilpotent trioid, the
free left (right) n-trinilpotent trioid, and the free rectangular trioid.
Some of these results can be applied to constructing relatively free
trialgebras.
1. Introduction
An associative trialgebra (an associative trioid) is a vector space (a set,
respectively) equipped with three binary associative operations ⊣, ⊢, and
⊥ satisfying the following eight axioms:
(x ⊣ y) ⊣ z = x ⊣ (y ⊢ z), (T1)
(x ⊢ y) ⊣ z = x ⊢ (y ⊣ z), (T2)
(x ⊣ y) ⊢ z = x ⊢ (y ⊢ z), (T3)
(x ⊣ y) ⊣ z = x ⊣ (y ⊥ z), (T4)
(x⊥ y) ⊣ z = x⊥ (y ⊣ z), (T5)
∗The author is supported by the National Research Foundation of Ukraine (grant
no. 2020.02/0066).
2020 MSC: 08B20, 20M10, 20M50, 17A30, 17D99.
Key words and phrases: trioid, trialgebra, free trioid, free trialgebra, relatively
free trioid, semigroup.
https://doi.org/10.12958/adm1732
“adm-n1” — 2021/4/10 — 20:38 — page 153 — #157
A. V. Zhuchok 153
(x ⊣ y)⊥ z = x⊥ (y ⊢ z), (T6)
(x ⊢ y)⊥ z = x ⊢ (y ⊥ z), (T7)
(x⊥ y) ⊢ z = x ⊢ (y ⊢ z). (T8)
In this paper, for short, we will use the terms “trialgebra” and “trioid”
to refer to an associative trialgebra and an associative trioid, respectively.
Trialgebras appeared first in the paper of Loday and Ronco [6] as
a non-commutative version of Poisson algebras. The operad associated
with trialgebras is Koszul dual to the operad associated with dendriform
trialgebras [6]. Trialgebras appeared to be naturally related to several
areas such as algebraic K-theory, Leibniz algebra theory, dialgebra theory,
dimonoid theory and semigroup theory. They are linear analogues of trioids
introduced also in [6]. If all operations of a trialgebra (trioid) coincide,
we obtain the notion of an associative algebra (semigroup), and if two
concrete operations of a trialgebra (trioid) coincide, we obtain the notion
of a dialgebra (dimonoid) and so, trialgebras (trioids) are a generalization
of associative algebras (semigroups) and dialgebras (dimonoids). Trioid
theory found some applications in trialgebra theory (see, e.g., [1–4]). Trioids
are also associated with n-tuple semigroups [13,15,22,26], dimonoids [5,9,
10,12,14,17,19,32] and g-dimonoids [7, 25,31,33]. Now trioid theory are
developed quite intensively. On one hand, some analogues of important
results from semigroup theory were proven. On the other hand, natural
questions about the structure of trioids are not considered. For example,
till now, free idempotent trioids, free products of trioids were not described.
In this survey, we want to gather and to systematize the main results
that belong to the variety theory of trioids and trialgebras. We will not
touch issues related to the study of congruences on trioids, we will not
focus on the properties and connections of trialgebras, as well as issues
related to the applications of trialgebras. Note, that most of the results
obtained to date relate to the theory of free systems in a trioid variety. It
happened thanks to the fact that relatively free objects in any variety of
algebras are important in the study of that variety. We will focus on the
results clarifying the structure of free objects in the varieties of trioids
and trialgebras. Our goal is to see which free objects have already been
constructed, and this will allow us to see which free objects should be
constructed further. First we give examples of trioids. Then recall and
summarize the results obtained by Loday and Ronco, the author as well
as others on the structure of free objects in the varieties of trialgebras and
trioids, namely, we give explicit structure theorems for the free trialgebra
and the free trioid, the free commutative trioid, the free n-nilpotent trioid,
“adm-n1” — 2021/4/10 — 20:38 — page 154 — #158
154 Structure of relatively free trioids
the free left (right) n-trinilpotent trioid, and the free rectangular trioid.
These results develop the variety theory of algebraic structures and some
of them can be applied to constructing relatively free trialgebras.
2. Examples of trioids
First we give examples of trioids.
a) Let S be a semigroup. A transformation ϕ of S is called an averaging
operator on S if ϕ is an endomorphism of S and
(xϕ y)ϕ = (x(yϕ))ϕ = xϕ yϕ
for all x, y ∈ S. A transformation ψ of a trioid (T,⊣,⊢,⊥) is called an
averaging operator on (T,⊣,⊢,⊥) [21] if ψ is an averaging operator on
(T,⊣), (T,⊢) and (T,⊥).
Let f be an idempotent endomorphism of S. Define operations ⊣, ⊢,
and ⊥ on S by
x ⊣ y = x(yf), x ⊢ y = (xf)y, x⊥ y = (xy)f
for all x, y ∈ S.
Proposition 2.1 ([21] , Proposition 3.1). (S,⊣,⊢,⊥) is a trioid.
The obtained trioid is denoted by Sf . It is known [21] that f is an
averaging operator on Sf .
b) Let S be a semigroup, and let f be an averaging operator on S.
Define operations ⊣, ⊢, and ⊥ on S by
x ⊣ y = x(yf), x ⊢ y = (xf)y, x⊥ y = xy
for all x, y ∈ S.
Proposition 2.2 ([21], Proposition 3.2). (S,⊣,⊢,⊥) is a trioid.
c) A semigroup S is called rectangular [21] if xyz = xz for all x, y,
z ∈ S.
Let S be a rectangular semigroup, let M be an arbitrary semigroup,
and let π :M → S be a homomorphism. Define operations ⊣, ⊢, and ⊥
on S ×M by
(s, t) ⊣ (p, g) = (s, tg), (s, t) ⊢ (p, g) = ((tπ)p, tg),
(s, t)⊥ (p, g) = (sp, tg)
for all (s, t), (p, g) ∈ S ×M .
“adm-n1” — 2021/4/10 — 20:38 — page 155 — #159
A. V. Zhuchok 155
Proposition 2.3 ([18], Proposition 8). (S ×M,⊣,⊢,⊥) is a trioid.
d) Let X be an arbitrary nonempty set, and let X∗ be the set of all
finite nonempty words over X. If w ∈ X∗, then denote the first (the last)
letter of a word w by w(0) (w(1), respectively). Define operations ⊣, ⊢,
and ⊥ on X∗ by
w ⊣ u = w(0)w(1), w ⊢ u = u(0)u(1), w ⊥ u = w(0)u(1)
for all w, u ∈ X∗.
Proposition 2.4 ([21], Proposition 3.4). (X∗,⊣,⊢,⊥) is a trioid.
Other numerous examples of trioids can be found in [6, 18,21,27–30].
3. Free trialgebras
The material of this section is based on [6]. Here we present the
constructions of the free trialgebra of an arbitrary rank and the free
trialgebra (trioid) of rank 1.
Let [n − 1] = {0, . . . , n− 1} be a set with n elements. The set of
nonempty subsets of [n− 1] is denoted by Pn. Observe that Pn is graded
by cardinality of its members. The subset of Pn whose members have
cardinality k is denoted by Pn,k. So, Pn = Pn,1 ∪ . . . ∪ Pn,n.
By definition, the free trialgebra over the vector space V is a trialgebra
Trias(V ) equipped with a map V → Trias(V ) which satisfies the following
universal property. For any map V → A, where A is a trialgebra, there is
a unique extension Trias(V ) → A which is a morphism of trialgebras.
The tensor product of vector spaces over a field K is denoted by ⊗.
Since the operations have no symmetry and since the relations let the
variables in the same order, Trias(V ) is completely determined by the free
trialgebra on one generator (i.e., V = K). The latter is a graded vector
space of the form
Trias(K) =
⊕
n>1
Trias(n).
From the motivation of defining the trialgebra type it is clear that for
n ∈ {1, 2, 3}, a basis of Trias(n) is given by the elements of P1, P2 and P3,
respectively.
Consider the bijection
bij : [i1 − 1] ∪ . . . ∪ [in − 1] → [i1 + . . .+ in − 1]
which sends k ∈ [ij − 1] to i1 + . . .+ ij−1 + k ∈ [i1 + . . .+ in − 1].
“adm-n1” — 2021/4/10 — 20:38 — page 156 — #160
156 Structure of relatively free trioids
Theorem 3.1 ([6], Theorem 1.7). The free trialgebra Trias(K) on one
generator is
⊕
n>1K[Pn] as a vector space. The binary operations ⊣, ⊥,
and ⊢ from K[Pp]⊗K[Pq] to K[Pp+q] are given by
X ⊣ Y = bij(X), X ⊥ Y = bij(X ∪ Y ), X ⊢ Y = bij(Y ),
where X ∈ Pp, Y ∈ Pq and bij : [p− 1]× [q − 1] → [p+ q − 1].
Corollary 3.2 ([6], Corollary 1.8). The free trialgebra Trias(V ) on the
vector space V is
Trias(V ) =
⊕
n>1
K[Pn]⊗ V ⊗n,
and the operations are induced by the operations on Trias(K) and concate-
nation.
Proposition 3.3 ([6], Proposition 1.9). The free trioid T on one generator
x is isomorphic to the trioid P =
⋃
n>1 Pn equipped with the operations
described in Theorem 3.1 above.
Since T is the free trioid generated by x, there exists a unique trioid
morphism φ : T → P .
Lemma 3.4 ([6], Lemma 1.10). Any complete parenthesizing of
(x ⊢ . . . ⊢ x)︸ ︷︷ ︸
a0
⊢ (x ⊣ . . . ⊣ x)︸ ︷︷ ︸
a1
⊥ (x ⊣ . . . ⊣ x)︸ ︷︷ ︸
a2
⊥ . . .⊥ (x ⊣ . . . ⊣ x)︸ ︷︷ ︸
ak
,
where a0 > 0, ai > 1 for i = 1, . . . , k, gives the same element, denoted ω,
in T. It is called the normal form of ω. Its image under φ in P is
x . . . x︸ ︷︷ ︸
a0
x . . . x︸ ︷︷ ︸
a1
x . . . x︸ ︷︷ ︸
a2
. . . x x . . . x︸ ︷︷ ︸
ak
.
4. Free trioids
Loday and Ronco constructed the free trioid of rank 1 [6] (see also
section 3). In this section, we describe the free trioid of an arbitrary rank
and for this trioid give an isomorphic construction. We also consider an
alternative construction for free trioids of rank 1.
Let X be an arbitrary nonempty set, X = {x | x ∈ X}, and let F[X]
be the free semigroup on X. Let further P ⊂ F[X ∪X] be a subsemigroup
which contains words w with the element x (x ∈ X) occuring in w at
“adm-n1” — 2021/4/10 — 20:38 — page 157 — #161
A. V. Zhuchok 157
least one time. For every w ∈ P denote by w̃ the word obtained from w
by change of all letters x (x ∈ X) by x. For instance, if w = xx yxz, then
w̃ = xxyxz. Obviously, w̃ ∈ F[X ∪X] \ P .
Define operations ⊣, ⊢, and ⊥ on P by
w ⊣ u = wũ, w ⊢ u = w̃u, w ⊥ u = wu
for all w, u ∈ P . The algebra (P,⊣,⊢,⊥) is denoted by Frt(X).
The proof of the following statement is the same as the proof of
Proposition 1.9 from [6] (see also Proposition 3.3) obtained for the free
trioid of rank 1.
Proposition 4.1 ([28], Proposition 1). Frt(X) is the free trioid.
If X = {x}, then Frt(X) is the free trioid of rank 1 presented by Loday
and Ronco in [6]. In the latter paper it was shown that the free trialgebra
over a vector space is completely determined by the free trialgebra on one
generator and the description of the latter trialgebra is reduced to the
description of the free trioid of rank 1 (see section 3).
Now we give another representation of the free trioid of an arbitrary
rank.
As usual,N denotes the set of all positive integers. For every word ω over
X the length of ω is denoted by ℓω. For any n, k ∈ N and L ⊆ {1, 2, . . . , n},
L 6= ∅, we let L+ k = {m+ k |m ∈ L}. Define operations ⊣′, ⊢′, and ⊥′
on the set
F = {(w,L) | w ∈ F[X], L ⊆ {1, 2, . . . , ℓw}, L 6= ∅}
by
(w,L) ⊣′ (u,R) = (wu,L), (w,L) ⊢′ (u,R) = (wu,R+ ℓw),
(w,L)⊥′ (u,R) = (wu,L ∪ (R+ ℓw))
for all (w,L), (u,R) ∈ F . The algebra (F,⊣′,⊢′,⊥′) is denoted by FT(X).
Theorem 4.2 ([21], Theorem 7.1). The free trioid Frt(X) is isomorphic
to the trioid FT(X).
Define operations ≺, ≻, and ↑ on the set
F ′ = {(n, L) | n ∈ N, L ⊆ {1, 2, . . . , n}, L 6= ∅}
by
(n, L) ≺ (m,R) = (n+m,L), (n, L) ≻ (m,R) = (n+m,R+ n),
(n, L) ↑ (m,R) = (n+m,L ∪ (R+ n))
for all (n, L), (m,R) ∈ F ′.
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158 Structure of relatively free trioids
From Theorem 4.2 we obtain
Corollary 4.3 ([21], Corollary 7.1). Let |X| = 1. The free trioid Frt(X)
of rank 1 is isomorphic to the trioid (F ′,≺,≻, ↑).
The trioid (F ′,≺,≻, ↑) was also considered in [35]. Another represen-
tation of the free trioid of rank 1 can be found in [16].
For free trioids, characterizations of the least left (right) zero con-
gruence, the least rectangular band congruence and the least n-nilpotent
congruence were given in [28] and [29], respectively. The description of
the least rectangular triband congruence on the free trioid follows from
Theorem 3.1 (i) in [28]. The problem of the characterization of the least
dimonoid congruences and the least semigroup congruence on the free
trioid was solved in [27]. The least commutative congruence, the least
commutative dimonoid congruences, and the least commutative semigroup
congruence on the free trioid were presented in [11]. Decompositions of
free trioids into tribands of subtrioids and bands of subtrioids were given
in [28]. In [34], it was proved that endomorphism semigroups of free trioids
are isomorphic if and only if the corresponding free trioids are isomorphic.
The endomorphism monoid of the free trioid of rank 1 was studied in [35].
5. Free commutative trioids
In this section, we construct the free commutative trioid of rank 1 and
show that the free commutative trioid of rank n > 1 is a subdirect product
of a free commutative semigroup of rank n and the free commutative
trioid of rank 1 [11]. We will use notations of section 4.
A trioid (T,⊣,⊢,⊥) is called commutative if semigroups (T,⊣), (T,⊢),
and (T,⊥) are commutative.
Let Ω be the free monoid on the 3-element set {a, b, c}, and let θ denote
the identity of Ω, that is, the empty word. By definition, the length ℓθ
of θ is equal to 0 and u0 = θ for any u ∈ Ω \ {θ}. For all u1, u2 ∈ Ω let
f⊣(u1, u2) = a, f⊢(u1, u2) =
{
b if u1 = u2 = θ,
a otherwise,
f⊥(u1, u2) =
{
c if u1 = ck, u2 = cp, k, p ∈ N ∪ {0},
a otherwise.
The subset
{yk | y ∈ {a, c}, k ∈ N ∪ {0}} ∪ {b}
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A. V. Zhuchok 159
of Ω is denoted by Ω. Define operations ⊣, ⊢, and ⊥ on Ω by
u1 ∗ u2 = f∗(u1, u2)
ℓu1+ℓu2+1
for all u1, u2 ∈ Ω and ∗ ∈ {⊣,⊢,⊥}. The algebra (Ω,⊣,⊢,⊥) is denoted
by FCT1.
Theorem 5.1 ([11], Theorem 3.1). FCT1 is the free commutative trioid
of rank 1.
Now we construct the free commutative trioid of an arbitrary rank.
Let X be an arbitrary nonempty set, and let F⋆[X] be the free com-
mutative semigroup on X. Define operations ⊣, ⊢, and ⊥ on the set
A = {(w, u) ∈ F⋆[X]× FCT1 | ℓw − ℓu = 1}
by
(w1, u1) ∗ (w2, u2) = (w1w2, f∗(u1, u2)
ℓu1+ℓu2+1)
for all (w1, u1), (w2, u2) ∈ A and ∗ ∈ {⊣,⊢,⊥}. The algebra (A,⊣,⊢,⊥)
is denoted by FCT(X).
Theorem 5.2 ([11], Theorem 3.8). FCT(X) is the free commutative
trioid.
A subdirect product of two algebras A1 and A2 is a subalgebra U of
the direct product A1 ×A2 such that the projection maps U → A1 and
U → A2 are surjections.
Corollary 5.3 ([11], Corollary 3.9). The free commutative trioid of rank
n > 1 is a subdirect product of a free commutative semigroup of rank n
and the free commutative trioid of rank 1.
Remark 5.4. From the construction of FCT(X) it follows that FCT(X)
is determined uniquely up to isomorphism by cardinality of the set X. Hence
the automorphism group of FCT(X) is isomorphic to the symmetric group
on X.
The least dimonoid congruences and the least semigroup congruence
on the free commutative trioid were described in [27].
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160 Structure of relatively free trioids
6. Free n-nilpotent trioids
In this section, we construct free n-nilpotent trioids of an arbitrary
rank and consider separately free n-nilpotent trioids of rank 1 [20,29]. We
will use notations of section 4.
An element 0 of a trioid (T,⊣,⊢,⊥) is called zero [21] if
x ∗ 0 = 0 ∗ x = 0 ∗ 0 = 0
for all x ∈ T and ∗ ∈ {⊣,⊢,⊥}. A trioid (T,⊣,⊢,⊥) with zero is called
nilpotent if for some n ∈ N and any xi ∈ T , 1 6 i 6 n + 1, and
∗j ∈ {⊣,⊢,⊥}, 1 6 j 6 n, any parenthesizing of
x1 ∗1 x2 ∗2 . . . ∗n xn+1
gives 0 ∈ T . The least such n is called the nilpotency index of (T,⊣,⊢,⊥).
For k ∈ N a nilpotent trioid of nilpotency index 6 k is said to be k-nil-
potent.
Fix n ∈ N. Let Pn ⊂ P be a set which contains words w with ℓw 6 n.
Define operations ≺,≻, and ↑ on the set Pn ∪ {0} by
w ≺ u =
{
wũ, ℓwu 6 n,
0, ℓwu > n,
w ≻ u =
{
w̃u, ℓwu 6 n,
0, ℓwu > n,
w ↑ u =
{
wu, ℓwu 6 n,
0, ℓwu > n,
w ∗ 0 = 0 ∗ w = 0 ∗ 0 = 0
for all w, u ∈ Pn and ∗ ∈ {≺,≻, ↑}. The algebra (Pn ∪ {0},≺,≻, ↑) is
denoted by P 0
n(X).
Theorem 6.1 ([29], Theorem 1). P 0
n(X) is the free n-nilpotent trioid.
Now we construct a trioid which is isomorphic to P 0
n(X).
Define operations ⊣, ⊢, and ⊥ on
FNTn = {(w,L) | w ∈ F[X], ℓw 6 n, L ⊆ {1, 2, . . . , ℓw}, L 6= ∅} ∪ {0}
by
(w,L) ⊣ (u,R) =
{
(wu,L), ℓwu 6 n,
0, ℓwu > n,
(w,L) ⊢ (u,R) =
{
(wu,R+ ℓw), ℓwu 6 n,
0, ℓwu > n,
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A. V. Zhuchok 161
(w,L)⊥ (u,R) =
{
(wu,L ∪ (R+ ℓw)), ℓwu 6 n,
0, ℓwu > n,
(w,L) ∗ 0 = 0 ∗ (w,L) = 0 ∗ 0 = 0
for all (w,L), (u,R) ∈ FNTn \{0} and ∗ ∈ {⊣,⊢,⊥}. The algebra
(FNTn,⊣,⊢,⊥) is denoted by FNTn(X).
Lemma 6.2 ([20], Lemma 1). FNTn(X) is a trioid.
Theorem 6.3 ([20], Theorem 3). The free n-nilpotent trioid P 0
n(X) is
isomorphic to the trioid FNTn(X).
Consider separately free n-nilpotent trioids of rank 1.
Define operations ⊣, ⊢, and ⊥ on the set
FNT′
n = {(m,L) | m ∈ N, m 6 n, L ⊆ {1, 2, . . . ,m}, L 6= ∅} ∪ {0}
by
(m,L) ⊣ (k,R) =
{
(m+ k, L), m+ k 6 n,
0, m+ k > n,
(m,L) ⊢ (k,R) =
{
(m+ k,R+m), m+ k 6 n,
0, m+ k > n,
(m,L)⊥ (k,R) =
{
(m+ k, L ∪ (R+m)), m+ k 6 n,
0, m+ k > n,
(m,L) ∗ 0 = 0 ∗ (m,L) = 0 ∗ 0 = 0
for all (m,L), (k,R) ∈ FNT′
n \{0} and ∗ ∈ {⊣,⊢,⊥}. The algebra
(FNT′
n,⊣,⊢,⊥) is denoted by FNT1
n.
Theorem 6.3 implies
Corollary 6.4 ([20], Corollary 1). If |X| = 1, then FNTn(X) ∼= FNT1
n.
Examples of nilpotent trioids of nilpotency index 2 can be found in [29].
Decompositions of free n-nilpotent trioids into 0-bands of subtrioids and
0-tribands of subtrioids were given in [29]. The least dimonoid congruences
and the least semigroup congruence on the free n-nilpotent trioid were
presented in [20].
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162 Structure of relatively free trioids
7. Free left (right) n-trinilpotent trioids
In this section, we construct the free left n-trinilpotent trioid and
consider free left n-trinilpotent trioids of rank 1 [23] separately. We will
use notations of section 4.
By Λ denote the signature of a trioid. Let a1, . . . , an be individual
variables. By P (a1, . . . , an) we denote the set of all terms of algebras of
the signature Λ having the form a1 ◦1 . . . ◦n−1 an with parenthesizing,
where ◦1, . . . , ◦n−1 ∈ Λ. A trioid (T,⊣,⊢,⊥) is called left trinilpotent if
for some n ∈ N, any a ∈ T and any p(a1, . . . , an) ∈ P (a1, . . . , an) the
following identities hold:
p(a1, . . . , an) ∗ a = p(a1, . . . , an), (7.1)
p(a1, . . . , an) ⊢ a = a1 ⊢ . . . ⊢ an, (7.2)
where ∗ ∈ {⊣,⊥}. The least such n is called the left trinilpotency index of
(T,⊣,⊢,⊥). For k ∈ N a left trinilpotent trioid of left trinilpotency index
6 k is said to be left k-trinilpotent. Obviously, in any trioid (T,⊣,⊢,⊥),
by axioms (T3), (T8) and associativity of the operation ⊢, we have
p(a1, . . . , an) ⊢ a = a1 ⊢ . . . ⊢ an ⊢ a.
Hence, if (T,⊢) is a left nilpotent semigroup of rank n [8], we get the
identity (7.2). This explains how we obtain the third identity in the
definition of a left trinilpotent trioid. Right k-trinilpotent trioids are
defined dually.
Let n, k ∈ N and L ⊆ {1, 2, . . . , n}. We regard
L+ k = {m+ k | m ∈ L}.
It is clear that ∅+k = ∅. For L 6= ∅, we let Lk,n = {m ∈ L | k+m 6 n},
and denote the least number of L by Lmin. Obviously,Lk,n = ∅ if k+m > n
for all m ∈ L.
Fix n ∈ N. Let w ∈ F[X]. If ℓw > n, let
n
−→w denote the initial subword
with the length n of w, and if ℓw < n, let
n
−→w= w. Define operations ⊣,⊢,
and ⊥ on
Vn = {(w,L) | w ∈ F[X], ℓw 6 n, L ⊆ {1, 2, . . . , ℓw}, L 6= ∅}
by
(w,L) ⊣ (u,R) = (
n
−→wu,L),
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A. V. Zhuchok 163
(w,L) ⊢ (u,R) =
(
n
−→wu, {n}), n < ℓw +Rmin,
(
n
−→wu,Rℓw,n + ℓw) otherwise,
(w,L)⊥ (u,R) = (
n
−→wu,L ∪ (Rℓw,n + ℓw))
for all (w,L), (u,R) ∈ Vn. The algebra (Vn,⊣,⊢,⊥) is denoted by FTl
n(X).
Theorem 7.1 ([23], Theorem 3.1). FTl
n(X) is the free left n-trinilpotent
trioid.
At the end of this section we construct a trioid which is isomorphic to
the free left n-trinilpotent trioid of rank 1.
Fix n ∈ N. For any m ∈ N let
n
−→m=
{
m, m 6 n,
n, m > n.
Define operations ⊣, ⊢, and ⊥ on
Mn = {(k, L) | k ∈ N, k 6 n, L ⊆ {1, 2, . . . , k}, L 6= ∅}
by
(k1, L) ⊣ (k2, R) = (
n
−−−−→
k1 + k2, L),
(k1, L) ⊢ (k2, R) =
(n, {n}), n < k1 +Rmin,
(
n
−−−−→
k1 + k2, R
k1,n + k1) otherwise,
(k1, L)⊥ (k2, R) = (
n
−−−−→
k1 + k2, L ∪ (Rk1,n + k1))
for all (k1, L), (k2, R) ∈Mn. The algebra (Mn,⊣,⊢,⊥) is denoted by F1T
l
n.
Theorem 7.1 implies the following statement.
Corollary 7.2 ([23], Corollary 3.11). If |X| = 1, then FTl
n(X) ∼= F1T
l
n.
Remark 7.3. In order to construct free right n-trinilpotent trioids we
use the duality principle.
It is known that the automorphism group of the free left (right) n-tri-
nilpotent trioid is isomorphic to the symmetric group [23]. The problem
of the description of the least left (right) n-trinilpotent congruence on the
free trioid was first announced in [24].
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164 Structure of relatively free trioids
8. Free rectangular trioids
In this section, we construct the free rectangular trioid [30].
A semigroup is called a left (right) zero semigroup provided that it
satisfies the identity xy = x (xy = y). A semigroup S is a rectangular band
if xyx = x for all x, y ∈ S. Equivalently, a semigroup S is a rectangular
band if x2 = x, xyz = xz for all x, y, z ∈ S. It is well-known that every
rectangular band is isomorphic to the Cartesian product of the left zero
semigroup and of the right zero semigroup. A trioid (T,⊣,⊢,⊥) is called
a rectangular trioid or a rectangular triband if (T,⊣), (T,⊢), and (T,⊥)
are rectangular bands.
Let X be an arbitrary nonempty set and X4 = X×X×X×X. Define
operations ⊣, ⊢, and ⊥ on X4 by
(x1, x2, x3, x4) ⊣ (y1, y2, y3, y4) = (x1, x2, x3, y4),
(x1, x2, x3, x4) ⊢ (y1, y2, y3, y4) = (x1, y2, y3, y4),
(x1, x2, x3, x4)⊥ (y1, y2, y3, y4) = (x1, x2, y3, y4)
for all (x1, x2, x3, x4), (y1, y2, y3, y4) ∈ X4. The algebra (X4,⊣,⊢,⊥) is
denoted by FRT(X).
Theorem 8.1 ([30], Theorem 1). FRT(X) is the free rectangular triband.
Examples of rectangular tribands can be found in [30]. Decompositions
of free rectangular tribands into bands of subtrioids, tribands of subsemi-
groups and tribands of subtrioids were given in [30]. It is known that the
automorphism group of FRT(X) is isomorphic to the symmetric group
on X, and any rectangular triband is semilattice indecomposable [30].
The least left (right) zero congruence and the least rectangular band
congruence on the free rectangular trioid were described in [30]. The least
dimonoid congruences and the least semigroup congruence on the free
rectangular trioid were presented in [27].
Note that the main results of sections 5–8 can be applied to constructing
the corresponding relatively free trialgebras.
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Contact information
Anatolii V.
Zhuchok
Department of Algebra and System Analysis,
Luhansk Taras Shevchenko National University,
Gogol square, 1, Starobilsk 92703, Ukraine
E-Mail(s): zhuchok.av@gmail.com
Received by the editors: 30.11.2020.
A. V. Zhuchok
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