Hyperstructures were introduced in 1934 when Marty [1] defined hypergroups, began to study their properties, and applied them to groups. Several papers and books have been written on hyperstructure theory [2,3]. Currently, a book published on hyperstructures [4] describes its applications in rough set theory, cryptography, automata, code automata, probability, geometry, lattices, binary relations, graphs, and hypergraphs.

Kazim and Naseeruddin [5] introduced the study of left almost semigroups (LA-semigroups). They generalized some useful sequels of semigroup theory. Subsequently, Mushtaq and his colleagues [6–13] further described the structure and included many useful results of theory of an LA-semigroup [7,14–20]. An LA-semigroup is the midway structure between a commutative semigroup and a groupoid. Moreover, it has many interesting properties, which are frequently used in commutative and associative algebraic structures.

Hila and Dine [21] presented the concept of LA-semihypergroups. Their paper created a new research direction in non-associative hyperstructures. Yaqoob et al. [22] extended the work of Hila and Dine [21] and characterized intra-regular LA-semihypergroups by their hyperideals using pure left identities. Subsequently, many researchers studied the structures of LA-semihypergroups from different perspectives [23–30].

Azhar and his colleagues [31–34] developed the structures of partially ordered LA-semihypergroups. Nawaz and his colleagues [35–41] developed and studied many non-associative structures. The concept of almost ideals in semigroups was provided by Grosek and Satko [42,43] in 1980 and 1981, respectively. Various other researchers studied almost ideals for different algebraic structures [44–46].

Recently, for the first time, Suebsung et. al [48] developed the concept of almost hyperideals in semihypergroups (see also [48,49]).

Thus, inspired by the concept by Suebsung et. al [47], we developed structures of almost hyperideals in LA-semihypergroups. These new types of almost hyperideals have two main characteristics. First, they are purely non-associative substructures of LA-semihypergroups. Second, in the construction of these hyperideals, we ensured that a hyperoperation did not yield a single element as in the scenario of the paper by Suebsung et. al [47]. We proved some useful results related to almost hyperideals.

In this section, we recall some basic concepts from the literature on ideals and LA-semihypergroups that were used in the further development of this article.

Definition 1

A map ◦ : X × X –→ ℘^*(X) is called an hyperoperation or join operation on the set X, where X is a non-empty set and ℘^*(X) = ℘(X)\{∅} indicates the all non-empty subsets of X. A hypergroupoid is a set (X) for which the binary operation is a hyperoperation.

If A and B are two non-empty subsets of X, then we express the product as follows:

Definition 2 ([21,22])

A hypergroupoid (X, ◦) is called an LA-semihypergroup if for x₁, x₂, x₃ ∈ X,

The law (x₁ ◦x₂)◦x₃ = (x₃ ◦x₂)◦x₁ is known as left invertive law.

Every LA-semihypergroup satisfies the law

for all x₁, x₂, x₃, x₄ ∈ X. This law is known as medial law (cf. [21]).

Definition 3. [22]

Let X be an LA-semihypergroup. Element e ∈ X is called a

1. Left identity (resp. pure left identity) if for all x₁ ∈ X, x₁ ∈ e ◦ x₁ (resp. x₁ = e ◦ x₁),

2. Right identity (resp. pure right identity) if for all x₁ ∈ X, x₁ ∈ x₁ ◦ e (resp. z₁ = z₁ ◦ e),

3. Identity (resp. pure identity) if for all z₁ ∈ H, z₁ ∈ e ◦ z₁ ∩ z₁ ◦ e (resp. x₁ = e ◦ z₁ ∩ z₁ ◦ e).

Example 1 ([22])

Let X = {l₁, l₂, l₃, l₄} with the binary hyperoperation defined as follows:

◦	l1	l2	l3	l4
l1	l1	l2	l3	l4
l2	l3	{l2, l3}	{l2, l3}	l4
l3	l2	{l2, l3}	{l2, l3}	l4
l4	l4	l4	l4	H

Hence X is an LA-semihypergroup because it satisfies the left invertive law. AS the table shows, l₁ is a left identity of X.

Lemma 1 ([22])

Let X be an LA-semihypergroup with pure left identity e; then x₁ ◦ (x₂ ◦ x₃) = x₂ ◦ (x₁ ◦ x₃) holds for all x₁, x₂, x₃ ∈ X.

Lemma 2 ([22])

Let X be an LA-semihypergroup with pure left identity e; then (x₁ ◦ x₂) ◦ (x₃ ◦ x₄) = (x₄ ◦ x₂) ◦ (x₃ ◦ x₁) holds for all x₁, x₂, x₃, x₄ ∈ X.

The law (x₁ ◦x₂)◦(x₃ ◦x₄) = (x₄ ◦x₂)◦(x₃ ◦x₁) is called a paramedial law.

Definition 4 ([47])

Let (H, ◦) be a semihypergroup.

(1) A non-empty subset L of H is called the left almost hyperideal of H if

(2) A non-empty subset L of H is called the right almost hyperideal of H if

(3) A non-empty subset L of H is called the almost hyper-ideal of H if L is both a left and right almost hyperideal of H.

Here we define left almost hyperideals, right almost hyperideals, almost hyperideals, minimal almost hyperideals, and some interesting properties.

Definition 5

Let (H, *) be an LA-semihypergroup.

(1) A non-empty subset L of H is called the left almost hyperideal of H if

(2) A non-empty subset R of H is called the right almost hyperideal of H if

(3) A non-empty subset I ofH is called the almost hyperideal of H if I is both a left and right almost hyperideal of H.

Example 2

Let H = {e, l, m} with the binary operation * defined as follows:

*	e	l	m
e	e	l	m
l	m	{l, m}	m
m	l	l	{l, m}

(H, *) is an LA-semihypergroup. Now, if L = {e, l} ⊆ H with we observe that (L, *) is a left almost hyperideal of (H, *), but it is not a right almost hyperideal as {e, l} * m ∩ {e, l} = φ.

*	e	l
e	e	l
l	m	{l, m}

Remark 1

The structure of left almost hyperideals is not LA-subsemihypergroups as in Example 2, where L = {e, l} ⊆ H is not closed under the same binary operation as H.

Example 3

Consider an LA-semihypergroup H = {a, b, c} under the hyperoperation defined by

◦	a	b	c
a	a	b	c
b	c	{b, c}	{b, c}
c	b	{b, c}	{b, c}

If R = {b, c} ⊆ H with the same binary operation ◦ that is defined in H, we can observe that R is a right almost hyperideal of H.

Lemma 3

Let (H, *) be a LA-semihypergroup.

(1) Every left hyperideal of H is a left almost hyperideal of H.

(2) Every right hyperideal of H is a left almost hyperideal of H.

(3) Every hyperideal of H is an almost hyperideal of H.

Remark 2

The converse of Lemma 3 may or may not true generally as in the following Example 4.

Example 4

Let H = {x, y, z} be a finite set with hyperoperation * defined as follows:

*	x	y	z
x	x	x	x
y	x	H	z
z	x	H	H

Thus, (H, *) is an LA-semihypergroup. Let L = {x, y} under the same binary operation *, i.e.,

*	x	y
x	x	x
y	x	H

Here, L is an almost hyperideal of H, but it is not a hyperideal of H.

Theorem 1

Let (H, *) be an LA-semihypergroup.

(1) If L₁ and L₂ are two left almost hyperideals of H, then L₁ ∪ L₂ is also a left almost hyperideal.

(2) If R₁ and R₂ are two right almost hyperideals of H, then R₁ ∪ R₂ is also a right almost hyperideal.

(3) If I₁ and I₂ are two almost hyperideals of H, then I₁ ∪I₂ is also an almost hyperideal.

Remark 3

(1) If L₁ and L₂ are two left almost hyperideals of H, then L₁ ∩ L₂ may or may not a left almost hyperideal.

(2) If R₁ and R₂ are two right almost hyperideals of H, then R ∩ R₂ may or may not a right almost hyperideal.

(3) If I₁ and I₂ are two almost hyperideals of H, then I₁ ∩I₂ may or may not left almost hyperideal.

Example 5

Let H = {1, 2, 3, 4, 5, 6} be a non-empty set with hyperoperation *, defined as follows: then (H, *) is an LA-semihypergroup. Let L₁ = {1, 2, 3, 4} and L₂ = {3, 4, 5, 6} be two left almost hyperideals of H. Thus, L₁ ∩ L₂ = {3, 4} is not a left almost hyperideal of H, as we can observe that 4 * {3, 4} ∩ {3, 4} = φ. However, L₁ ∪ L₂ = {1, 2, 3, 4, 5, 6} is a left almost hyperideal of H.

*	1	2	3	4	5	6
1	{1, 2, 3, 4}	{2, 3, 4, 5, 6}	{4, 5, 6}	{2, 3, 4, 5, 6}	{2, 3, 4, 5, 6}	{2, 3, 4, 5, 6}
2	{1, 3, 4, 5, 6}	{1, 2, 3, 4, 5, 6}	{1, 3, 4, 5, 6}	{1, 3, 6}	{1, 5, 6}	{1, 3, 4, 5, 6}
3	{1, 2, 4, 5, 6}	{1, 2}	{1, 2, 3, 5, 6}	{1, 2, 4, 5, 6}	{1, 2, 6}	{1, 2, 4, 5, 6}
4	{1, 5, 6}	{1, 2, 6}	{1, 2, 5, 6}	{1, 2, 5, 6}	{1, 2, 5, 6}	{1, 2, 5, 6}
5	{1, 2, 3, 4, 6}	{1, 2, 3, 4, 6}	{1, 2, 3, 4, 6}	{1, 3, 4, 6}	{1, 2, 3, 4, 5, 6}	{1, 3, 4, 6}
6	{1, 2, 3, 4, 5}	{1, 4, 5}	{1, 2, 3, 4, 5}	{1, 4, 5}	{1, 2, 3, 4, 5}	{1, 2, 3, 4, 5}

Lemma 4

If L is a left almost hyperideal of an LA-semihyper-group H with a left identity e; then, aL is a left almost hyper-ideal of H.

Lemma 5

If L is a left almost hyperideal of an LA-semihypergroupH with a left identity e, then L₂ is a left almost hyperideal of H.

Lemma 6

If I is a left almost hyperideal of H, then I₂ is an almost hyperideal of H.

Definition 6

An almost hyperideal I of an LA-semihypergroup H is minimal if it does not contain any almost hyperideal of H other than itself.

Example 6

Let H = {x₁, x₂, x₃, x₄} with the binary hyperoperation defined as follows:

◦	x1	x2	x3	x4
x1	x1	x2	x3	x4
x2	x3	{x2, x3}	{x2, x3}	x4
x3	x2	{x2, x3}	{x2, x3}	x4
x4	x4	x4	x4	H

Hence H is an LA-semihypergroup. Let I₁ = {x₁, x₂, x₄}, I₂ = {x₁, x₃, x₄} and I₃ = {x₄} are the almost hyperideals of H. Thus, I₃ is a minimal almost hyperideal of H.

Theorem 2

Let H be an LA-semihypergroup. If L is a minimal almost left hyperideal of H, then a*L is also a minimal almost left hyperideal of H for every weak idempotent a.

In this paper, we have introduced new types of hyperideals in non-associative structures of semihypergroups and indicated some interesting properties. In the future, we aim to obtain more properties of the proposed ideal theory.