Epigenetic Inheritance via Fuzzy Soft Hyperstructures

1. Introduction

Algebraic hyperstructures connect abstract algebra with biological systems. Marty [5] introduced hyperstructures where the composition of two elements yields a set [2]. This approach fits genetic crosses, where mating results in a set of distinct offspring genotypes or phenotypes.

Ghadiri et al. [4] showed that the mating operation on the $F_2$ generation of a gene pool forms an $H_v$-semigroup. Vougiouklis [8] and Davvaz [3] analyzed these weak hyperstructures. Al-Tahan and Davvaz [1] extended this framework to n-hybrid crosses and non-Mendelian inheritance, including epistasis. Preethi et al. [7] integrated fuzzy sets to model inheritance on fuzzy hyperlattice ordered groups.

Standard algebraic models usually treat genetic outcomes as static, relying only on DNA sequences. However, epigenetic mechanisms show that environmental factors alter gene expression without changing the underlying nucleotide order. External triggers, such as thermal stress, soil nutrient variations, and noise, alter the fixed composition tables of classical hyperstructures. To model these variations, we embed Molodtsov’s soft set theory [6] into weak hyperstructures. This approach constructs parameter-dependent algebraic operations that map genotype crossings onto fuzzy soft $H_v$-semigroups.

The rest of this paper is organized as follows. Section 2 reviews the mathematical preliminaries, establishing the groundwork for the fuzzy soft $H_v$-semigroups introduced in Section 3. Section 4 develops the parameterized genetic model, which is analyzed through a biological case study in Section 5. Finally, Section 6 provides concluding remarks and future research directions.

2. Mathematical Preliminaries

Let H be a non-empty set, and let ${\mathcal{P}}^{*}\left(H\right)$ be the collection of its non-empty subsets.

Definition 1.

A mapping $\circ :H\times H\to {\mathcal{P}}^{*}\left(H\right)$ is a hyperoperation on H. The pair $(H,\circ )$ is a hypergroupoid.

For non-empty subsets $A,B\subseteq H$ and $x\in H$, we define:

$$A\circ B=\bigcup _{a\in A,b\in B}a\circ b,x\circ A=\left\{x\right\}\circ A,A\circ x=A\circ \left\{x\right\}.$$

Definition 2.

A hypergroupoid $(H,\circ )$ is an $H_v$-semigroup if the weak associativity axiom holds:

$$(x\circ (y\circ z\left)\right)\cap \left(\right(x\circ y)\circ z)\ne \varnothing ,\forall x,y,z\in H.$$

An element $e\in H$ is an identity of $(H,\circ )$ if $x\in (x\circ e)\cap (e\circ x)$ for each $x\in H$. Let $\mathcal{F}\left(H\right)$ be the collection of all fuzzy subsets on H, defined by mappings $\mu :H\to [0,1]$.

Definition 3.

A mapping $\circ :H\times H\to \mathcal{F}\left(H\right)$ is a fuzzy hyperoperation on H. The pair $(H,\circ )$ is a fuzzy hypergroupoid.

The fuzzy hyperproduct $(\mu \circ \nu )\in \mathcal{F}\left(H\right)$ of $\mu ,\nu \in \mathcal{F}\left(H\right)$ is given for each $z\in H$ by:

$$(\mu \circ \nu )\left(z\right)=\underset{x,y\in H}{sup}\{\mu \left(x\right)\wedge (x\circ y)\left(z\right)\wedge \nu \left(y\right)\}.$$

Let U be the universe set and E the parameter space.

Definition 4.

A pair $(F,E)$ is a soft set over U if F is a mapping $F:E\to \mathcal{P}\left(U\right)$.

Definition 5.

A pair $(G,E)$ is a fuzzy soft set over U if G is a mapping $G:E\to \mathcal{F}\left(U\right)$.

3. Soft $H_v$-Semigroups

Let H be a non-empty set, let E be a parameter set, and let $\mathcal{F}\left(H\right)$ be the collection of all fuzzy subsets on H.

Definition 6.

A mapping ${\circ }_E:H\times H\to \mathcal{F}{\left(H\right)}^E$ is a fuzzy soft hyperoperation on H with respect to the parameter set E. For $x,y\in H$ and $e\in E$, the output $\left(x{\circ }_{E}y\right)\left(e\right)$ is a fuzzy subset of H with the membership function ${\mu }_{\left(x{\circ }_{E}y\right)\left(e\right)}:H\to [0,1]$.

If A is a fuzzy soft set over H parametrized by E, the component fuzzy set for a parameter $e\in E$ is $A\left(e\right)$ with the membership function ${\mu }_{A\left(e\right)}$. For $x\in H$, the fuzzy soft hyperproduct of an element and a fuzzy soft set under parameter $e\in E$ is defined for each $z\in H$ by:

$$\left(x{\circ }_{E}A\right)\left(e\right)\left(z\right)=\underset{y\in H}{sup}\{{\mu }_{\left(x{\circ }_{E}y\right)\left(e\right)}\left(z\right)\wedge {\mu }_{A\left(e\right)}\left(y\right)\}$$

$$\left(A{\circ }_{E}x\right)\left(e\right)\left(z\right)=\underset{y\in H}{sup}\{{\mu }_{\left(y{\circ }_{E}x\right)\left(e\right)}\left(z\right)\wedge {\mu }_{A\left(e\right)}\left(y\right)\}$$

For two fuzzy soft sets A and B, the hyperproduct under parameter $e\in E$ is defined for each $z\in H$ by:

$$\left(A{\circ }_{E}B\right)\left(e\right)\left(z\right)=\underset{x,y\in H}{sup}\{{\mu }_{A\left(e\right)}\left(x\right)\wedge {\mu }_{\left(x{\circ }_{E}y\right)\left(e\right)}\left(z\right)\wedge {\mu }_{B\left(e\right)}\left(y\right)\}$$

Definition 7.

The triple $(H,{\circ }_E,E)$ is a fuzzy soft $H_v$-semigroup if the weak associativity axiom holds, meaning that for all $x,y,z\in H$ and $e\in E$, there exists $t\in H$ such that:

$${\mu }_{\left[x{\circ }_E\left(y{\circ }_{E}z\right)\right]\left(e\right)}\left(t\right)>0\mathrm{and}{\mu }_{\left[\left(x{\circ }_{E}y\right){\circ }_{E}z\right]\left(e\right)}\left(t\right)>0.$$

Definition 8.

Let $(H,{\circ }_E,E)$ be a fuzzy soft $H_v$-semigroup. An element $u\in H$ is a soft identity element if, for all $x\in H$ and $e\in E$:

$${\mu }_{\left(x{\circ }_{E}u\right)\left(e\right)}\left(x\right)>0\mathrm{and}{\mu }_{\left(u{\circ }_{E}x\right)\left(e\right)}\left(x\right)>0.$$

Definition 9.

Let $(H,{\circ }_E,E)$ be a fuzzy soft $H_v$-semigroup. A non-empty subset $K\subseteq H$ is a fuzzy soft $H_v$-subsemigroup of H if $(K,{\circ }_E,E)$ is a fuzzy soft $H_v$-semigroup.

Theorem 1.

Let $(H,{\circ }_E,E)$ be a fuzzy soft $H_v$-semigroup and let $K\subseteq H$. If for all $x,y,z\in K$ and $e\in E$, there exists $t\in K$ such that ${\mu }_{\left[x{\circ }_E\left(y{\circ }_{E}z\right)\right]\left(e\right)}\left(t\right)>0$ and ${\mu }_{\left[\left(x{\circ }_{E}y\right){\circ }_{E}z\right]\left(e\right)}\left(t\right)>0$, then K is a fuzzy soft $H_v$-subsemigroup of H.

Proof. 

Let $x,y,z\in K\subseteq H$ and $e\in E$. Choose an element $t\in K$ for which

$${\mu }_{\left[x{\circ }_E\left(y{\circ }_{E}z\right)\right]\left(e\right)}\left(t\right)>0\mathrm{and}{\mu }_{\left[\left(x{\circ }_{E}y\right){\circ }_{E}z\right]\left(e\right)}\left(t\right)>0$$

Weak associativity holds, and the triple $(K,{\circ }_E,E)$ inherits the properties of a fuzzy soft $H_v$-semigroup. □

Proposition 1.

Let $(H,{\circ }_E,E)$ be a fuzzy soft $H_v$-semigroup. If ${\circ }_E$ is a crisp soft hyperoperation, where ${\mu }_{\left(x{\circ }_{E}y\right)\left(e\right)}\left(z\right)\in \{0,1\}$ for all $x,y,z\in H$ and $e\in E$, then $(H,{\circ }_E,E)$ reduces to a classical soft $H_v$-semigroup.

Proof. 

Binary membership values in $\{0,1\}$ restrict the infimum (∧) operator to a classical intersection (∩). Under this constraint, the condition

$${\mu }_{\left[x{\circ }_E\left(y{\circ }_{E}z\right)\right]\left(e\right)}\left(t\right)>0\wedge {\mu }_{\left[\left(x{\circ }_{E}y\right){\circ }_{E}z\right]\left(e\right)}\left(t\right)>0$$

is equivalent to the crisp set intersection:

$$\left(x{\circ }_E\left(y{\circ }_{E}z\right)\right)\left(e\right)\cap \left(\left(x{\circ }_{E}y\right){\circ }_{E}z\right)\left(e\right)\ne \varnothing .$$

□

4. Algebraic Modeling of Epigenetic Inheritance

Let G be the set of genotypes in the $F_2$ generation from a monohybrid cross. Following the framework of Ghadiri et al., we consider the gene pool $\{A,\overline{A},a,\overline{a}\}$ containing the dominant alleles $A,\overline{A}$ and the recessive alleles $a,\overline{a}$. The resulting collection of $F_2$ genotypes forms our biological universe:

$$G=\{AA,A\overline{A},Aa,A\overline{a},\overline{A}\overline{A},\overline{A}a,\overline{A}\overline{a},aa,a\overline{a},\overline{a}\overline{a}\}.$$

Let $E=\{e_1,e_2,e_3,e_4\}$ be the set of external environmental parameters:

• $e_1$: Control environment, regulated by traditional Mendelian genetic hyperoperations.
• $e_2$: High thermal stress, which uses DNA methylation to silence genes and suppress dominant traits.
• $e_3$: Methyl-donor rich soil environment that uses chromatin remodeling to make latent alleles more likely to be expressed.
• $e_4$: Acoustic stress, which triggers physiological responses that shift gene expression.

We model epigenetic mating as a fuzzy soft hyperoperation ${\circ }_E:G\times G\to \mathcal{F}{\left(G\right)}^E$. Given a parameter $e\in E$ and any two genotypes $X,Y\in G$, the resulting cross $\left(X{\circ }_{E}Y\right)\left(e\right)$ is a fuzzy subset of G, evaluated by the membership function ${\mu }_{\left(X{\circ }_{E}Y\right)\left(e\right)}:G\to [0,1]$.

• Case 1: Standard Environment ($e_1$)
  Without external environmental factors, the membership values of classical genetic crosses belong to the binary set $\{0,1\}$:

  $${\mu }_{\left(X{\circ }_{E}Y\right)\left(e_1\right)}\left(Z\right)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}Z\in X\times Y\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$
  The cross $Aa\times Aa$ yields:

  $$\left(Aa{\circ }_{E}Aa\right)\left(e_1\right)=\{AA/1.0,Aa/1.0,aa/1.0\}.$$
• Case 2: High Thermal Stress ($e_2$)
  Temperatures above 30°C trigger epigenetic gene silencing and reduce the expression of the dominant allele A. For an $Aa\times Aa$ cross, this effect shifts the membership degrees toward the recessive spectrum:

  $$\left(Aa{\circ }_{E}Aa\right)\left(e_2\right)=\{AA/0.4,Aa/0.8,aa/1.0\}.$$
  The dominant genotype $AA$ has a lower membership value, while the recessive genotype $aa$ reaches full membership.
• Case 3: Methyl-Donor Rich Substrate ($e_3$)
  Substrates rich in methyl donors, such as soil supplemented with choline or folate, alter gene structures. These chemical conditions stabilize heterozygous forms, increasing the hybrid membership degrees during an $Aa\times Aa$ cross:

  $$\left(Aa{\circ }_{E}Aa\right)\left(e_3\right)=\{AA/0.9,Aa/1.0,aa/0.3\}.$$
• Case 4: Acoustic Stress ($e_4$)
  Loud noise alters DNA methylation patterns and suppresses dominant traits. This constraint changes the output distribution:

  $$\left(Aa{\circ }_{E}Aa\right)\left(e_4\right)=\{AA/0.5,Aa/0.7,aa/1.0\}.$$
  Noise reduces the presence of dominant traits, leaving the recessive genotype $aa$ with full membership.

Environmental variations change the offspring distribution probabilities, but the underlying algebraic structure remains invariant. The proposed model incorporates non-Mendelian patterns, such as epistasis and codominance, using the parameterized hyperoperation:

$${\circ }_E:G\times G\times E\to \mathcal{F}\left(G\right)$$

5. Case Study

We provide a biological verification based on temperature-dependent flower pigmentation to evaluate weak associativity.

Let the snapdragon (Antirrhinum) flower phenotype space be defined over the genotype subspace $G^{\prime }=\{AA,Aa,aa\}$. Allele A produces red pigment and allele a results in white flowers, while the heterozygotes ($Aa$) show pink or red phenotypes depending on the dominance mechanisms.

We analyze an epigenetic triplet cross with the parental components $X=AA$, $Y=Aa$, and $Z=aa$. Weak associativity is tested under four conditions: the control environment ($e_1$), thermal stress ($e_2$), methyl-donor rich soil ($e_3$), and acoustic stress ($e_4$).

Evaluation under Standard Environment ($e_1$)

From the composition laws in Table 1, the initial hyperproduct is:

$$\left(AA{\circ }_{E}Aa\right)\left(e_1\right)=\{AA/1.0,Aa/1.0\}$$

For the left-hand path $\left(X{\circ }_{E}Y\right){\circ }_{E}Z$, taking the union of the components results in:

$$\left[\left(AA{\circ }_{E}Aa\right){\circ }_{E}aa\right]\left(e_1\right)=\left(\{AA/1.0,Aa/1.0\}{\circ }_{E}aa\right)\left(e_1\right)=\{Aa/1.0,aa/1.0\}$$

Table 1. Fuzzy Soft Genetic Mating $\left(X{\circ }_{E}Y\right)\left(e\right)$ over Subspace $G^{\prime }$

${\circ }_{\mathit{E}}$	$\mathit{A}\mathit{A}$	$\mathit{A}\mathit{a}$	$\mathit{a}\mathit{a}$
Parameter $e_1$ (Standard)
$AA$	$\{AA/1.0\}$	$\{AA/1.0,Aa/1.0\}$	$\{Aa/1.0\}$
$Aa$	$\{AA/1.0,Aa/1.0\}$	$\{AA/1.0,Aa/1.0,aa/1.0\}$	$\{Aa/1.0,aa/1.0\}$
$aa$	$\{Aa/1.0\}$	$\{Aa/1.0,aa/1.0\}$	$\{aa/1.0\}$
Parameter $e_2$ (Thermal Stress)
$AA$	$\{AA/0.6,Aa/0.2\}$	$\{AA/0.5,Aa/0.7\}$	$\{AA/0.6,aa/0.4\}$
$Aa$	$\{AA/0.5,Aa/0.7\}$	$\{AA/0.4,Aa/0.8,aa/1.0\}$	$\{Aa/0.5,aa/0.9\}$
$aa$	$\{AA/0.6,aa/0.4\}$	$\{Aa/0.5,aa/0.9\}$	$\{aa/1.0\}$
Parameter $e_3$ (Rich Soil)
$AA$	$\{AA/1.0\}$	$\{AA/0.9,Aa/1.0\}$	$\{Aa/1.0\}$
$Aa$	$\{AA/0.9,Aa/1.0\}$	$\{AA/0.9,Aa/1.0,aa/0.3\}$	$\{Aa/1.0,aa/0.4\}$
$aa$	$\{AA/1.0\}$	$\{Aa/1.0,aa/0.4\}$	$\{aa/0.8\}$
Parameter $e_4$ (Acoustic Stress)
$AA$	$\{AA/0.7,Aa/0.3\}$	$\{AA/0.6,Aa/0.6\}$	$\{AA/0.7,aa/0.3\}$
$Aa$	$\{AA/0.6,Aa/0.6\}$	$\{AA/0.5,Aa/0.7,aa/1.0\}$	$\{Aa/0.6,aa/0.8\}$
$aa$	$\{AA/0.7,aa/0.3\}$	$\{AA/0.6,aa/0.8\}$	$\{aa/1.0\}$

Table 1. Fuzzy Soft Genetic Mating $\left(X{\circ }_{E}Y\right)\left(e\right)$ over Subspace $G^{\prime }$

${\circ }_{\mathit{E}}$	$\mathit{A}\mathit{A}$	$\mathit{A}\mathit{a}$	$\mathit{a}\mathit{a}$
Parameter $e_1$ (Standard)
$AA$	$\{AA/1.0\}$	$\{AA/1.0,Aa/1.0\}$	$\{Aa/1.0\}$
$Aa$	$\{AA/1.0,Aa/1.0\}$	$\{AA/1.0,Aa/1.0,aa/1.0\}$	$\{Aa/1.0,aa/1.0\}$
$aa$	$\{Aa/1.0\}$	$\{Aa/1.0,aa/1.0\}$	$\{aa/1.0\}$
Parameter $e_2$ (Thermal Stress)
$AA$	$\{AA/0.6,Aa/0.2\}$	$\{AA/0.5,Aa/0.7\}$	$\{AA/0.6,aa/0.4\}$
$Aa$	$\{AA/0.5,Aa/0.7\}$	$\{AA/0.4,Aa/0.8,aa/1.0\}$	$\{Aa/0.5,aa/0.9\}$
$aa$	$\{AA/0.6,aa/0.4\}$	$\{Aa/0.5,aa/0.9\}$	$\{aa/1.0\}$
Parameter $e_3$ (Rich Soil)
$AA$	$\{AA/1.0\}$	$\{AA/0.9,Aa/1.0\}$	$\{Aa/1.0\}$
$Aa$	$\{AA/0.9,Aa/1.0\}$	$\{AA/0.9,Aa/1.0,aa/0.3\}$	$\{Aa/1.0,aa/0.4\}$
$aa$	$\{AA/1.0\}$	$\{Aa/1.0,aa/0.4\}$	$\{aa/0.8\}$
Parameter $e_4$ (Acoustic Stress)
$AA$	$\{AA/0.7,Aa/0.3\}$	$\{AA/0.6,Aa/0.6\}$	$\{AA/0.7,aa/0.3\}$
$Aa$	$\{AA/0.6,Aa/0.6\}$	$\{AA/0.5,Aa/0.7,aa/1.0\}$	$\{Aa/0.6,aa/0.8\}$
$aa$	$\{AA/0.7,aa/0.3\}$	$\{AA/0.6,aa/0.8\}$	$\{aa/1.0\}$

For the right-hand sequence $X{\circ }_E\left(Y{\circ }_{E}Z\right)$ we obtain:

$$\left(Aa{\circ }_{E}aa\right)\left(e_1\right)=\{Aa/1.0,aa/1.0\}$$

$$\left[AA{\circ }_E\left(Aa{\circ }_{E}aa\right)\right]\left(e_1\right)=\left(AA{\circ }_E\{Aa/1.0,aa/1.0\}\right)\left(e_1\right)=\{AA/1.0,Aa/1.0\}$$

The intersection of the respective support sets yields:

$$\mathrm{Supp}\left(\left[\left(AA{\circ }_{E}Aa\right){\circ }_{E}aa\right]\left(e_1\right)\right)\cap \mathrm{Supp}\left(\left[AA{\circ }_E\left(Aa{\circ }_{E}aa\right)\right]\left(e_1\right)\right)=\{Aa,aa\}\cap \{AA,Aa\}=\left\{Aa\right\}$$

Because the genotype $Aa$ remains present in both pathways with full membership weight ($1.0>0$), the weak associativity holds under $e_1$.

Evaluation under Thermal Stress Environment ($e_2$)

High temperatures lead to DNA methylation, which alters dominant expression profiles. Evaluating the left-hand path for the terminal element $t=aa$ via the supremum mapping gives:

$$\left(AA{\circ }_{E}Aa\right)\left(e_2\right)=\{AA/0.5,Aa/0.7\}$$

$$\left[\left(AA{\circ }_{E}Aa\right){\circ }_{E}aa\right]\left(e_2\right)\left(aa\right)=max\{\left(AA{\circ }_{E}aa\right)\left(e_2\right)\left(aa\right)\wedge 0.5,\left(Aa{\circ }_{E}aa\right)\left(e_2\right)\left(aa\right)\wedge 0.7\}$$

$$=max\{0.4\wedge 0.5,0.9\wedge 0.7\}=0.7$$

The right-hand path gives:

$$\left(Aa{\circ }_{E}aa\right)\left(e_2\right)=\{Aa/0.5,aa/0.9\}$$

$$\left[AA{\circ }_E\left(Aa{\circ }_{E}aa\right)\right]\left(e_2\right)\left(aa\right)=max\{\left(AA{\circ }_{E}Aa\right)\left(e_2\right)\left(aa\right)\wedge 0.5,\left(AA{\circ }_{E}aa\right)\left(e_2\right)\left(aa\right)\wedge 0.9\}$$

$$=max\{0.0\wedge 0.5,0.4\wedge 0.9\}=0.4$$

Evaluating both operational pathways for $t=aa$ shows that:

$$\left[AA{\circ }_E\left(Aa{\circ }_{E}aa\right)\right]\left(e_2\right)\left(aa\right)=0.4>0\mathrm{and}\left[\left(AA{\circ }_{E}Aa\right){\circ }_{E}aa\right]\left(e_2\right)\left(aa\right)=0.7>0$$

This positive intersection ensures that the weak associative law holds even under environmental stress vectors.

Evaluation under Methyl-Donor Rich Soil Environment ($e_3$)

Substrates rich in methyl donors stabilize heterozygous forms. To test weak associativity under $e_3$, we evaluate the genotype outcome for $t=Aa$. From Table 1, the left intermediate step is:

$$\left(AA{\circ }_{E}Aa\right)\left(e_3\right)=\{AA/0.9,Aa/1.0\}$$

Combining this result with the third genotype using the supremum composition yields:

$$\left[\left(AA{\circ }_{E}Aa\right){\circ }_{E}aa\right]\left(e_3\right)\left(Aa\right)=max\{\left(AA{\circ }_{E}aa\right)\left(e_3\right)\left(Aa\right)\wedge 0.9,\left(Aa{\circ }_{E}aa\right)\left(e_3\right)\left(Aa\right)\wedge 1.0\}.$$

$$=max\{1.0\wedge 0.9,1.0\wedge 1.0\}=1.0.$$

For the right pathway, Table 1 provides the second sequence:

$$\left(Aa{\circ }_{E}aa\right)\left(e_3\right)=\{Aa/1.0,aa/0.4\}.$$

Composing this outcome with the first parental component yields:

$$\left[AA{\circ }_E\left(Aa{\circ }_{E}aa\right)\right]\left(e_3\right)\left(Aa\right)=max\{\left(AA{\circ }_{E}Aa\right)\left(e_3\right)\left(Aa\right)\wedge 1.0,\left(AA{\circ }_{E}aa\right)\left(e_3\right)\left(Aa\right)\wedge 0.4\}.$$

$$=max\{1.0\wedge 1.0,1.0\wedge 0.4\}=1.0.$$

Evaluating both paths for the hybrid genotype $Aa$ shows that:

$$\left[AA{\circ }_E\left(Aa{\circ }_{E}aa\right)\right]\left(e_3\right)\left(Aa\right)=1.0>0\mathrm{and}\left[\left(AA{\circ }_{E}Aa\right){\circ }_{E}aa\right]\left(e_3\right)\left(Aa\right)=1.0>0.$$

The genotype $Aa$ maintains positive membership values in both cases, satisfying weak associativity.

Evaluation under Acoustic Stress Environment ($e_4$)

Acoustic stress induces physiological tension that shifts gene expression. We check weak associativity under noise for the element $t=aa$.

From Table 1, the left intermediate step is:

$$\left(AA{\circ }_{E}Aa\right)\left(e_4\right)=\{AA/0.6,Aa/0.6\}.$$

The supremum definition yields the value for the target genotype:

$$\left[\left(AA{\circ }_{E}Aa\right){\circ }_{E}aa\right]\left(e_4\right)\left(aa\right)=max\{\left(AA{\circ }_{E}aa\right)\left(e_4\right)\left(aa\right)\wedge 0.6,\left(Aa{\circ }_{E}aa\right)\left(e_4\right)\left(aa\right)\wedge 0.6\}$$

$$=max\{0.3\wedge 0.6,0.8\wedge 0.6\}=0.6.$$

The right pathway yields:

$$\left(Aa{\circ }_{E}aa\right)\left(e_4\right)=\{Aa/0.6,aa/0.8\}.$$

Combining this fuzzy set with the remaining component gives:

$$\left[AA{\circ }_E\left(Aa{\circ }_{E}aa\right)\right]\left(e_4\right)\left(aa\right)=max\{\left(AA{\circ }_{E}Aa\right)\left(e_4\right)\left(aa\right)\wedge 0.6,\left(AA{\circ }_{E}aa\right)\left(e_4\right)\left(aa\right)\wedge 0.8\}$$

$$=max\{0.6\wedge 0.6,0.3\wedge 0.8\}=0.6.$$

Both paths result in positive membership values for $aa$:

$$\left[AA{\circ }_E\left(Aa{\circ }_{E}aa\right)\right]\left(e_4\right)\left(aa\right)=0.6>0\mathrm{and}\left[\left(AA{\circ }_{E}Aa\right){\circ }_{E}aa\right]\left(e_4\right)\left(aa\right)=0.6>0.$$

Environmental parameters alter the algebraic properties of the system. Under standard control conditions ($e_1$), the operation satisfies the classical reproduction axiom $G{\circ }_{E}G=G$. Since weak associativity holds, the framework forms a genetic $H_v$-hypergroup.

Under environmental stress profiles ($e_2,e_3$, and $e_4$), strict associativity fails, meaning $X{\circ }_E\left(Y{\circ }_{E}Z\right)\ne \left(X{\circ }_{E}Y\right){\circ }_{E}Z$. The system then behaves as an $H_v$-semigroup where a non-empty intersection replaces set equality. These environmental changes also modify the identity properties, shifting the genotype $Aa$ from a strong identity element under $e_1$ to a weak fuzzy identity element under thermal stress ($e_2$).

6. Conclusions and Future Directions

This paper connects abstract algebra with epigenetics by introducing fuzzy soft $H_v$-semigroups. By applying soft sets to model genetic crosses, we show that weak associativity holds under environmental factors such as thermal stress, substrate nutrient variations, and acoustic noise.

Future research can expand this framework in two distinct directions. First, we plan to study ordered fuzzy soft hyperstructures to model gene dominance mechanisms. Second, we aim to develop rough soft hyperstructures to handle missing or incomplete data in experimental laboratories.