1. Introduction
A poset is a set with a binary relation satisfying reflexivity, antisymmetry and transitivity. Researches and generalizations on posets are very rich. The most famous result on posets is the decomposition theorem [1] proposed by Dilworth in 1950, also well-known as Dilworth’s Theorem, which has great combinatorial and order theoretical value. To learn more about Dilworth’s Theorem, please refer to Fulkerson [2], Tverberg [3], Pretzel [4] and Galvin [5].
In 1964, Rota [6] made the Möbius function emerge in clear view as a fundamental invariant, which unifies both enumerative and structural aspects of the theory of partially ordered sets. In 1972, Stanley [7] studied ordered structures and partitions. Later, he proved several identities associated with the binomial posets [8]. In 1977, Trotter and Moore [9] studied the dimension of planar posets and the dimension of trees. In 1988, Stanley [10] first introduced the differential poset with combinatorial and algebraic properties. For more works on differential posets, see [11] [12] [13] [14] [15].
In 2005, Aguiar and Sottile [16] introduced the global descents of permutations in the symmetric group
. In 2020, based on the global descents, Zhao and Li [17] studied a new shuffle product
on permutations. Later, they [18] defined a new product
and a new coproduct
on permutations, proved that
is a graded
-algebra and
is a graded
-coalgebra, where
is a field, and studied some properties of the structures. In 2021, Liu and Li [19] introduced the super-shuffle product and the cut-box coproduct on permutations and proved that
is a graded algebra and
is a graded coalgebra. These papers are helpful for us to study algebra and coalgebra on posets.
In 2020, Aval, Bergeron and Machacek [20] defined a product and a coproduct on posets without proofs. In this paper, we prove that the vector space spanned by posets with these operations is an algebra and a coalgebra, respectively.
We start by recalling some basic definitions of algebra and coalgebra and some notations on posets in Section 2. In Section 3, we introduce the definitions of the conjunction product and the unshuffle coproduct on the vector space spanned by posets. Then we prove the vector space with the conjunction product is a free graded algebra. And the vector space with the unshuffle coproduct is a graded coalgebra. Thus, we construct algebra and coalgebra structures on posets. Finally, we make a summary of this paper in Section 4.
2. Preliminaries
2.1. Basic Definitions
We recall some basic definitions of algebra and coalgebra; see [21] [22] for more details. Let
be an associative commutative ring with identity.
For an
-module
, we call
an
-algebra if there exist two maps
and
such that the diagrams in Figure 1 are commutative. Here
is called a product and
a unit.
The R-algebra
is graded if
and
, for all
.
We reverse all the arrows in Figure 1 to get the definition of coalgebra since algebra and coalgebra are dual concepts.
For an
-module
, we call
an
-coalgebra if there exist two maps
and
such that the diagrams in Figure 2 are commutative. Here
is called a coproduct and
a counit.
The
-coalgebra
is graded if
and
, for all
.
A coalgebra
is cocommutative if the diagram in Figure 3 commutes, where
, for all
in
.
Let
be a vector space over field
. Denote the tensor algebra on
by
, where
and
Define the multiplication on
by the concatenation product and unit
(a)
(b)
Figure 1. Associative law and unitary property. (a) Associative law; (b) Unitary property.
(a)
(b)
Figure 2. Coassociative law and counitary property. (a) Coassociative law; (b) Counitary property.
on
by
, for
. Then the algebra
is free on
since it satisfies the following universal property: for each
-algebra
and each linear map
, there exists a unique algebra homomorphism
such that
where
is the inclusion map.
2.2. Basic Notations
Now let’s recall some notations on posets; see [23] [24] for more details.
A partial order relation is a binary relation satisfying reflexivity, antisymmetry and transitivity. A set
together with a partial order relation
is called a poset, denoted by
. The set
is called the ground set of poset
. We denote the number of elements of
by
. When the ground set is empty, we have an empty poset, denoted by
. When the partial order relation is obvious,
can represent both the ground set and the poset.
For distinct elements
in poset
, if
and there is no element
that differs from
and satisfies
, then we say that
covers
, denoted by
, and we also call
a cover relation in
. Define
to be the set of all cover relations in
by
If
is given in poset
, then we can get the partial order relation
corresponding to the cover relations through reflexivity, antisymmetry and transitivity. Obviously,
and
are uniquely determined by each other.
The two elements
and
in poset
are called comparable, if either
or
. In a poset, it is not necessary that any two elements are comparable. When
and
are two elements of
such that neither
nor
, they are called incomparable.
To study posets more intuitively, we can represent posets by Hasse diagrams. A Hasse diagram is a graphical rendering of a poset displayed by the cover relations of the poset with an implied upward orientation. Drawing line segments between these elements of a poset follows these two rules:
1) If
in the poset, then the point representing
is lower in the drawing than the point representing
.
2) Drawing line segment between the points representing elements
and
of the poset if
covers
.
In addition, incomparable elements can be drawn on the same layer.
For example, for set
with
, i.e., 3 covers 2 and 4 covers 3, according to transitivity, we have
. Further, we get poset
. Similarly, if
with
, then poset
.
In poset
, an element
is called maximal in
, if there is no other element
in
satisfying
. Similarly,
is minimal if no other element
in
satisfing
. A partially ordered set may have more than one maximal or minimal elements. Hence, we denote
as the set containing all maximal elements in
and
containing all minimal elements in
. From the above example, we have
,
,
and
.
Let
and
be disjoint sets with partial orders
and
, respectively. Define
to be the union of
and
with partial order
given by the cover relations
We denote the poset
as
, which means
is an operation for posets. Obviously,
satisfies the associative law. For example, Let
and
, then
,
,
and
. Hence we have
, i.e.,
.
Denote
to be the set of all posets on
, where
. For example, when
,

Let
be the disjoint union of
, and
, where
is the linear space spanned by
over field
.
For a positive integer
, define
as a poset by increasing each element in
by
satisfying
for
in
. Similarly, define
as a poset by reducing each element in
by
satisfying
for
in
. For example, let
, then
and
.
For a poset
in
, we call
a global split of
if
where
. If a nonempty poset has no global splits except 0 and
, we call it an indecomposable poset. We denote
as the subset of
containing all indecomposable posets in
, and
.
For a nonempty set of intergers
with
, denote
as a mapping from
to
by
for each
.
Let
be a poset, where
is an interger set. We define
, where
and
is the partial order on
satisfying
for any
in
. For convenience, we denote the
as
when the partial order
is obvious. We call
the standard form of poset
. For example,
. Obviously,
.
Let
be a subset of
. Define
as poset
, where
is the partial order on
satisfying
for any
in
. For convenience, we denote
by
when the partial order
is obvious, and call
the restriction of
on
, and
a
subposet of poset
. For example, let poset
, then 
Obviously,
,
and
are subposets of poset
.
In particular,
and
, for
in
.
3. Conjunction Product and Unshuffle Coproduct
In 2020, Aval, Bergrron and Machacek [20] defined a product and coproduct on posets, which are called the conjunction product and the unshuffle coproduct, respectively, without proofs. Here, we prove that the vector space spanned by posets with these operations is an algebra and a coalgebra.
Define the conjunction product
on
by
for
in
and
in
. Obviously,
in
, and the conjunction product
is not commutive.
Define the unit
by
.
Example 1. Let
and
, then
![]()
Theorem 1.
is a graded algebra.
Proof It is easy to verify that
is a unit.
Let
be in
with
,
. By the definition of conjunction product
, we have
and
. Obviously,
is in
and
is in
. Furthermore,
and
are both in
. We have
From above,
satisfies the associative law. Hence,
is an algebra. From the definitions of conjunction product
and unit
, we have
and
. Hence, the algebra
is graded. ![]()
Lemma 1. Define a linear mapping
by
(1)
where
,
for
and
. Denote
For any
(2)
and
(3)
if
, then
.
Proof If
, then there exists
such that
for
but
. We denote
,
and
. If
, then
So
. If
, then
is a split of
but not a split of
. So
. Similarly, if
, we also have
. ![]()
Theorem 2. The algebra
is free on
.
Proof It is sufficient to prove
is isomorphic to the tensor algebra
through the mapping
in (1). Obviously,
is an algebra homomorphism. For any nonempty
in
, let
be all splits of
, then
Hence,
is surjective.
For
in
, where
and
for
,
, suppose
(4)
i.e.,
(5)
Obviously, any two terms in (5) are linearly independent because they have different numbers of splits. It means
and
(6)
for all
. By the associative law of tensor product, we have
for some
,
and
, where
for
. Then
(7)
By Lemma 1,
are linear independent. So
for all
from (6) and (7). Then
for all
. Hence,
in (4), i.e.,
is injective. Then
is a free algebra on
. ![]()
Define the unshuffle coproduct
on
by
where
traverses all subsets of
, for any non-empty poset
in
and
. Obviously, the unshuffle coproduct
is cocommutive. Define the counit
by
for
in
.
Example 2. Let
, then![]()
Theorem 3.
is a graded coalgebra.
Proof It is easy to verify that
is a counit. For
in
,
We have
and
Therefore
From above,
satisfies the coassociative law. Hence,
is a coalgebra.
From the definitions of
and
, we have
and
. Hence, the coalgebra
is graded. ![]()
4. Conclusions
Let
be the vector space spanned by posets. Firstly, we give the definitions of conjunction product
and unshuffle coproduct
on
. Then we prove that the conjunction product
satisfies the associativity. So
is an algebra. Futhermore, we prove that
is graded and free on
, where
contains all indecomposable posets in
. Finally, we prove that unshuffle coproduct
satisfies the coassociativity and
is a graded coalgebra.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Nos. 11701339, 12071265 and 11771256).