Matrix expression of finite BZMV dM -algebra

Abstract

In this paper, we provide a systematic characterization of finite BZMV^dM-algebra by using semi-tensor product of matrices. The abstract operation law about logic of the finite algebra is transformed into the simple operation of concrete logical matrices. In addition, we study some properties of BZMV^dM-algebra, such as homomorphism, isomorphism, and the product of the BZMV^dM-algebra. Through logical matrix operation, the direct verifiable conditions for detecting the above properties are given.

Keywords

semi-tensor product of matrices isomorphism

1 Introduction

BZMV^dM-algebra was firstly proposed by Gattaneo, Giuntini and Pilla in 1997 [1]. It is both an MV-algebra and a distributive de Morgan BZ-lattice. MV-algebra have been introduced by Chang [2] in order to provide an adequate semantic characterization for Lukasiewicz multi-valued logics. Brouwer-Zadeh lattices, first investigated in [3], give rise to fuzzy forms of quantum logics. The defination of BZMV^dM-algebra is completely equational, which will be very useful for application to multi-valued logics and their algebraic semanties. Gattaneo, Giuntini and Pilla [1] studied the structure and sketched the main properties of BZMV^dM-algebra and prove that BZMV^dM-algebra are same as the Stonian MV-algebra. Gattaneo and Ciucci [4] used BZMV^dM-algebra as an abstract environment to describe both shadowed and fuzzy set, in such structure it is possible to algebraically define an operator which given a fuzzy set returns a particular induced shadowed set. Gao, Deng and Liu [5] showed that BZMV^dM-algebra can induce a quasi-lattice implication algebra, FI algebra, lattice implication algebra and weak algebra.

In the process of concrete research on BZMV^dM-algebra, it is not easy to find the finite algebra satisfying the conditions, judge the homomorphism and isomorphism, and obtain the operator operation law of the product of two algebras. Therefore, by using the semi-tensor product (STP) of matrices, the abstract logic operation law is transformed into a simple operation of concrete logic matrix and the operators satisfying the conditions in finite BZMV^dM-algebra are characterized by structural matrix. STP was first proposed by researcher Daizhan Cheng as a new matrix product, which has been used in many fields, including complex k-valued logic operations, Boolean networks, fuzzy control, etc [6 –8]. It also can be applied for the analysis and control of dynamic systems [9], which can be used to represent the logical variables in a matrix form to realize logic reasoning processes. This paper concerns only finite BZMV^dM-algebra, the matrix expression of finite BZMV^dM-algebra will be proposed by using STP.

The main contribution of this paper include: (i) The BZMV^dM-algebra is systematically characterized by STP method; (ii) Using matrix expressions, the abstract operation law about logic is transformed into the simple operation of concrete logical matrices; (iii) The isomorphism classification problem of finite BZMV^dM-algebra is solved, which has not been solved completely before.

This paper is organized as follows. In Section 2, we recall STP and the matrix expression of k-valued logical functions which will be used in this paper. In Section 3, we consider the matrix expression of finite BZMV^dM-algebra. In Section 4, Using the structure matrices, the homomorphism and isomorphism of the BZMV^dM-algebras are investigated. In Section 5, the product of the algebra is considered. Finally, some conclusions are put in Section 6.

In this paper, we adopt the following notations.

• $ℝ^{n}$ stands for the set of all real column vectors with order n. $ℝ^{m \times n}$ stands for the set of all m × n real matrices.

•Col (A) (Row (A)) stand for the set of columns (rows) of A; Col_i (A) (Row_i (A)) stand for the i-th column (row) of matrix A.

•I_n represents the unit matrix with order n. $δ_{n}^{k}$ represents the k-th column of I_n.

•1_k represents the k-dimensional column vector whose elements are all one.

• $Δ_{n} : = {δ_{n}^{k} | k = 1, \dots, n} .$

• $D : = {0, 1} . D_{k}$ represents the set ${0, \frac{1}{k - 1}, \dots,$ $\frac{k - 2}{k - 1}, 1}$ , in particular $D_{2} = D = {0, 1}$ .

• $L_{n \times r} : = {L : L \in ℝ^{n \times r} and Col (L) \subset Δ_{n}}$ . And any matrix $L \in L_{n \times r}$ is called a logical matrix.

•If $L \in L_{n \times r}$ , then it is expressed and briefly denoted as $L = [δ_{n}^{i_{1}}, δ_{n}^{i_{2}}, \dots, δ_{n}^{i_{r}}]$ and L = δ_n [i₁, i₂, ⋯, i_r].

•The symbol ⋉ and ⊗ are the semi-tensor product and Kronecker product of matrices, respectively.

2 Preliminary

This section provides a brief survey on STP of matrices and the algebraic expression of k-valued logical functions. First, we give the definition of the STP of matrices as follows.

Definition 2.1. [6] Let $A \in ℝ^{m \times n}, B \in ℝ^{p \times q}, t = lcm (n, p)$ is the least common multiple of n and p. Then, the semi-tensor product of A and B is defined as $A ⋉ B : = (A \otimes I_{t / n}) (B \otimes I_{t / p}) .$

When n = p, A⋉B = AB. The semi-tensor product of matrices is a generalization of conventional matrix product.

In STP of matrices, it cannot achieve the general commutativity, but also has the following properties.

Proposition 2.2. [6] Let $X, Y^{T} \in ℝ^{t}$ , for any arbitrary real matrix A, we have $X ⋉ A = (I_{t} \otimes A) ⋉ X,$ $A ⋉ Y = Y ⋉ (I_{t} \otimes A) .$

Proposition 2.3. [11] Let $X \in ℝ^{m}, Y \in ℝ^{n}$ , then $W_{[m, n]} ⋉ X ⋉ Y = Y ⋉ X,$ in which $\begin{matrix} W_{[m, n]} = & [I_{n} \otimes δ_{m}^{1}, I_{n} \otimes δ_{m}^{2}, \dots, I_{n} \otimes δ_{m}^{m}] \\ \in ℝ^{mn \times mn} \end{matrix}$ is called swap matrix.

Proposition 2.4. [11] The swap matrix is invertible and $W_{[m, n]}^{T} = W_{[m, n]}^{- 1} = W_{[n, m]} .$

In order to use STP of matrices to deal with logical problems, it is a key to express logical relations in matrix form. First, we use vectors to represent logical variables.

Definition 2.5. [12] Suppose $X \in D$ , then the vector form expression of $X$ is denoted as $\vec{X}$ $x = \vec{X} : = [\begin{matrix} X \\ 1 - X \end{matrix}], X \in D$

Similar to classical logic, the vector form expression can also be used at multi-valued logic. Consider k-valued logic, let $X = \frac{i}{k - 1} \in D_{k}, i = 0, 1, \dots, k - 1$ . $x = \vec{X} = \vec{(\frac{i}{k - 1})} : = δ_{k}^{k - i}, i = 0, 1, \dots, k - 1 .$

Under the vector form expression, k-valued logical variables x ∈ Δ_k.

Lemma 2.6. [12] Let F: ${D_{k}}^{n} \to D_{k}$ be an n-ary k-valued logical function. Using vector form expression, F can be expressed as f: Δ_kⁿ → Δ_k. There exists a unique logical matrix $M_{F} \in L_{k \times k^{n}}$ , M_F is the structure matrix of F, such that $f (x_{1}, \dots, x_{n}) = M_{F} ⋉_{i = 1}^{n} x_{i}, x_{i} \in Δ_{k},$ in which $⋉_{i = 1}^{n} x_{i} = x_{1} ⋉ x_{2} ⋉ \dots ⋉ x_{n}$ .

Example 2.7. Considering 3-valued logical (k=3), let $ξ \in D_{3} = {1, \frac{1}{2}, 0}$ , so the vector form expression $x = \vec{ξ} \in$ Δ₃. In other word, $\vec{1} = δ_{3}^{1}, \vec{\frac{1}{2}} = δ_{3}^{2}, \vec{0} = δ_{3}^{3}$ . Let $ξ \in D_{3} = {1, \frac{1}{2}, 0}$ .

Consider the logical operator negation “¬”, ¬ξ = 1 - ξ. Denote the structure matrice of “¬” as M_n. By Lemma 2.6, $\begin{matrix} \neg \vec{1} = M_{n} ⋉ δ_{3}^{1} = {Col}_{1} (M_{n}) = δ_{3}^{3}, \\ \neg \vec{(\frac{1}{2})} = M_{n} ⋉ δ_{3}^{2} = {Col}_{2} (M_{n}) = δ_{3}^{2}, \\ \neg \vec{0} = M_{n} ⋉ δ_{3}^{3} = {Col}_{3} (M_{n}) = δ_{3}^{1} . \end{matrix}$ It follows that M_n = δ₃ [3, 2, 1].

Proposition 2.8. [13] Suppose x ∈ Δ_k, the power reducing matrix is defined as ${PR}_{k} = diag (δ_{k}^{1}, δ_{k}^{2}, \dots, δ_{k}^{k}), k = 2, 3, \dots$ , then we have the following formula, $x^{2} = {PR}_{k} x .$

Remark 2.9. In this paper, the traditional matrix product is assumed to be STP, so the symbol “⋉” is mostly omitted for convenience. Supplementary information of properties about STP of matrices is included in the Appendix.

3 Matrix expression of finite BZMV^dM-algebra

In this section, we introduce the BZMV^dM-algebra and give the matrix expression of finite BZMV^dM-algebra.

Definition 3.1. [1] A BZMV^dM-algebra is a system 〈A, ⊕, ¬, ∼, 0〉, where A is a non-empty set of elements, 0 is a constant element of A, ¬ and ∼ are unary operations on A, ⊕ is a binary operation on A, obeying the following axioms:

(x ⊕ y) ⊕ z = (y ⊕ z) ⊕ x,

x ⊕ 0 = x,

¬ (¬ x) = x,

¬ (¬ x ⊕ y) ⊕ y = ¬ (x ⊕ ¬ y) ⊕ x,

∼x ⊕ ∼∼x = ¬0,

x ⊕ ∼∼x = ∼∼x,

∼¬ [¬ (x ⊕ ¬ y) ⊕ ¬ y] = ¬ (∼∼x ⊕ ¬ ∼∼y) ⊕¬∼∼y.

Consider a finite set A = {b₁, b₂, ⋯, b_k}, k < ∞. We convert the elements of A into vector form as $\vec{b_{1}} = δ_{k}^{1}, \vec{b_{2}} = δ_{k}^{2}, \dots, \vec{b_{k}} = δ_{k}^{k} (b_{k} : = 0)$ . Assume M_⊕ (k), M_n (k) and M_B (k) (briefly denoted by M_⊕, M_n and M_B) are structure matrices of ⊕, ¬ and ∼, respectively, then we have the following new definition of equivalent algebraic conditions for BZMV^dM-algebra.

Definition 3.2. Assume |A| = k < ∞, 〈A, ⊕, ¬, ∼, 0〉 is a BZMV^dM-algebra, obeying the following axioms,

M_⊕² = M_⊕²W_[k²,k],

$M_{\oplus} W_{[k, k]} δ_{k}^{k} = I_{k},$

M_n² = I_k,

(M_⊕M_n) ² (I_k ⊗ PR_k)

= M_⊕M_nM_⊕ (I_k ⊗ M_nW_[k,k]) PR_k,

$M_{\oplus} M_{B} (I_{k} \otimes {M_{B}}^{2}) {PR}_{k} = 1_{k}^{T} \otimes (M_{n} δ_{k}^{k}),$

M_⊕ (I_k ⊗ M_B²) PR_k = M_B²,

$M_{B} (M_{n} M_{\oplus})^{2} (I_{k} \otimes M_{n}) (I_{k}^{2} \otimes M_{n}) (I_{k} \otimes {PR}_{k}) = M_{\oplus} M_{n} M_{\oplus} {M_{B}}^{2} (I_{k} \otimes M_{n} M_{B}^{2}) (I_{k^{2}} \otimes M_{n} M_{B}^{2})$ (I_k ⊗ PR_k).

Proof. We only prove (4), other items can be proved similarly. Expressing ¬ (¬ x ⊕ y) ⊕ y = ¬ (x ⊕ ¬ y) ⊕ x into algebraic form, we have $M_{\oplus} (M_{n} (M_{\oplus} (M_{n} xy)) y) = M_{\oplus} (M_{n} (M_{\oplus} ({xM}_{n} y)) x),$ (1) according to Proposition 2.2 and 2.8, the left hand side of Equation (1) is $\begin{matrix} {(M_{\oplus} M_{n})}^{2} {xy}^{2} & = {(M_{\oplus} M_{n})}^{2} x {PR}_{k} y \\ = {(M_{\oplus} M_{n})}^{2} (I_{k} \otimes {PR}_{k}) xy . \end{matrix}$ By Proposition 2.2, 2.3, 2.8 and 7.2, the right hand side of Equation (1) is $\begin{matrix} M_{\oplus} M_{n} M_{\oplus} {xM}_{n} yx \\ = M_{\oplus} M_{n} M_{\oplus} (I_{k} \otimes M_{n}) xyx \\ = M_{\oplus} M_{n} M_{\oplus} (I_{k} \otimes M_{n}) {xW}_{[k, k]} xy \\ = M_{\oplus} M_{n} M_{\oplus} (I_{k} \otimes M_{n}) (I_{k} \otimes W_{[k, k]}) {PR}_{k} xy \\ = M_{\oplus} M_{n} M_{\oplus} (I_{k} \otimes M_{n} W_{[k, k]}) {PR}_{k} xy . \end{matrix}$ Since x, y, z ∈ Δ_k are arbitrary, (4) follows.□

Through the conditions of the Definition 3.2, we can construct the structure matrix of BZMV^dM-algebra. The following example is used to illustrate the construction of the structure matrix.

Example 3.3. (i) Let k = 2, M_⊕ (2) = δ₂ [a₁, a₂, a₃, a₄], M_n (2) = δ₂ [b₁, b₂], M_B (2) = δ₂ [c₁, c₂], according to (2) of Definition 3.2, we have a₂ = 1, a₄ = 2. Then $M_{\oplus} (2) = δ_{2} [a_{1}, 1, a_{3}, 2] .$

According to (3) of Definition 3.2, we have b₁ ≠ b₂. There are two Kleene orthocomplementations “ ¬ ″, which are $M_{n}^{1} (2) = δ_{2} [1, 2], M_{n}^{2} (2) = δ_{2} [2, 1] .$ The unary operation “∼” have four possibilities, which are $M_{B}^{1} = δ_{2} [1, 1], M_{B}^{2} = δ_{2} [1, 2],$ $M_{B}^{3} = δ_{2} [2, 1], M_{B}^{4} = δ_{2} [2, 2] .$

Using the exhaustive method to test, it easy to figure out that there only one BZMV^dM-algebra follows the algebraic conditions in Definition 3.2, which is

M_{\oplus} (2) = δ_{2} [1, 1, 1, 2],

(2)

M_{n} (2) = δ_{2} [2, 1],

(3)

M_{B} (2) = δ_{2} [2, 1] .

(4) The BZMV^dM-algebra 〈 {a, 0}, ⊕, ¬, ∼, 0〉 is characterized by the tables:

Table 1

	¬	∼
a	0	0
0	a	a

Table 2

		⊕
a	a	a
a	0	a
0	a	a
0	0	0

(ii) Let k = 3, similar calculations show that there are two BZMV^dM-algebras, which are

\begin{matrix} M_{\oplus}^{1} (3) = δ_{3} [1, 1, 1, 1, 1, 2, 1, 2, 3], \\ M_{n}^{1} (3) = δ_{3} [3, 2, 1], M_{B}^{1} (3) = δ_{3} [3, 3, 1], \end{matrix}

(5) and

\begin{matrix} M_{\oplus}^{2} (3) = δ_{3} [2, 2, 1, 2, 2, 2, 1, 2, 3], \\ M_{n}^{2} (3) = δ_{3} [1, 3, 2], M_{B}^{2} (3) = δ_{3} [3, 3, 2] . \end{matrix}

(6)

Let A = {a, b, 0},

\vec{a} = δ_{3}^{1}, \vec{b} = δ_{3}^{2}, \vec{0} = δ_{3}^{3}

. The BZMV^dM-algebra 〈A, ⊕ _i, ¬ _i, _∼i, 0〉, i = 1, 2 is characterized by the tables:

Table 3

	¬₁	∼₁	¬₂	∼₂
a	0	0	a	0
b	b	0	0	0
0	a	a	b	b

Table 4

		⊕₁	⊕₂
a	a	a	b
a	b	a	b
a	0	a	a
b	a	a	b
b	b	a	b
b	0	b	b
0	a	a	a
0	b	b	b
0	0	0	0

(iii) Let k = 4, similar calculations show as follows:

$\begin{matrix} M_{\oplus}^{1} (4) = δ_{4} [1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 3, 4], \\ M_{n}^{1} (4) = δ_{4} [4, 3, 2, 1], M_{B}^{1} (4) = δ_{4} [4, 4, 4, 1] . \end{matrix}$ $\begin{matrix} M_{\oplus}^{2} (4) = δ_{4} [1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 3, 1, 2, 3, 4], \\ M_{n}^{2} (4) = δ_{4} [4, 3, 2, 1], M_{B}^{2} (4) = δ_{4} [4, 3, 2, 1] . \end{matrix}$ $\begin{matrix} M_{\oplus}^{3} (4) = δ_{4} [1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 2, 3, 4], \\ M_{n}^{3} (4) = δ_{4} [4, 3, 2, 1], M_{B}^{3} (4) = δ_{4} [4, 4, 4, 1] . \end{matrix}$ $\begin{matrix} M_{\oplus}^{4} (4) = δ_{4} [1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 3, 1, 2, 3, 4], \\ M_{n}^{4} (4) = δ_{4} [3, 4, 1, 2], M_{B}^{4} (4) = δ_{4} [3, 4, 1, 2] . \end{matrix}$ $\begin{matrix} M_{\oplus}^{5} (4) = δ_{4} [1, 3, 3, 1, 3, 2, 3, 2, 3, 3, 3, 3, 1, 2, 3, 4], \\ M_{n}^{5} (4) = δ_{4} [2, 1, 4, 3], M_{B}^{5} (4) = δ_{4} [2, 1, 4, 3] . \end{matrix}$ $\begin{matrix} M_{\oplus}^{6} (4) = δ_{4} [2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 3, 1, 2, 3, 4], \\ M_{n}^{6} (4) = δ_{4} [3, 4, 1, 2], M_{B}^{6} (4) = δ_{4} [4, 4, 4, 2] . \end{matrix}$ $\begin{matrix} M_{\oplus}^{7} (4) = δ_{4} [2, 3, 3, 1, 3, 3, 3, 2, 3, 3, 3, 3, 1, 2, 3, 4], \\ M_{n}^{7} (4) = δ_{4} [2, 1, 4, 3], M_{B}^{7} (4) = δ_{4} [4, 4, 4, 3] . \end{matrix}$ $\begin{matrix} M_{\oplus}^{8} (4) = δ_{4} [3, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 4], \\ M_{n}^{8} (4) = δ_{4} [3, 4, 1, 2], M_{B}^{8} (4) = δ_{4} [4, 4, 4, 2] . \end{matrix}$ $\begin{matrix} M_{\oplus}^{9} (4) = δ_{4} [3, 3, 3, 1, 3, 1, 3, 2, 3, 3, 3, 3, 1, 2, 3, 4], \\ M_{n}^{9} (4) = δ_{4} [2, 1, 4, 3], M_{B}^{9} (4) = δ_{4} [4, 4, 4, 3] . \end{matrix}$

Namely, when k = 4, there are nine kinds of BZMV^dM algebras satisfying Definition 3.2.

Using BZMV^dM-algebra structure, we define several new operations: $\begin{matrix} 1 : = \neg 0, \\ x ⊙ y : = \neg (\neg x \oplus \neg y), \\ x \lor y : = (x ⊙ \neg y) \oplus y = \neg (\neg x \oplus y) \oplus y, \\ x \land y : = (x \oplus \neg y) ⊙ y = \neg [\neg (x \oplus \neg y) \oplus \neg y] . \end{matrix}$

Assume M_⊙, M_d and M_c are structure matrices of ⊙, ∨ and ∧, respectively. Similarly, the matrix expression is $1 : = M_{n} δ_{k}^{k} = {Col}_{k} (M_{n}),$ (7) $M_{⊙} = M_{n} M_{\oplus} M_{n} (I_{k} \otimes M_{n}),$ (8) $M_{d} = (M_{\oplus} M_{n})^{2} (I_{k} \otimes {PR}_{k}),$ (9) $\begin{matrix} M_{c} & = & (M_{n} M_{\oplus})^{2} (I_{k} \otimes M_{n}) (I_{k^{2}} \otimes M_{n}) (I_{k} \otimes {PR}_{k}) . \end{matrix}$ (10)

Example 3.4. Let A = {1, a, 0}, $\vec{1} = δ_{3}^{1}, \vec{a} = δ_{3}^{2}, \vec{0} = δ_{3}^{3}$ . According to (ii) in Example 3.3, the structure matrix of operators ⊕, ¬, ∼, ⊙, ∨ and ∧ are expressed as follows: $\begin{matrix} M_{\oplus} = δ_{3} [1, 1, 1, 1, 1, 2, 1, 2, 3], \\ M_{n} = δ_{3} [3, 2, 1], M_{B} = δ_{3} [3, 3, 1] . \end{matrix}$ Using Equation (8)–(10), we have $\begin{matrix} M_{⊙} = δ_{3} [3, 3, 3, 3, 3, 2, 3, 2, 1], \\ M_{d} = δ_{3} [1, 1, 1, 1, 2, 2, 1, 2, 3], \\ M_{c} = δ_{3} [1, 2, 3, 2, 2, 3, 3, 3, 3] . \end{matrix}$

The properties satisfied by the BZMV^dM-algebra can be transformed into the corresponding algebraic form through matrix expression, as follows.

Proposition 3.5. If A is a BZMV^dM-algebra, M_⊕, M_n and M_B are structure matrices of ⊕, ¬ and ∼, then the following results are true:

M_⊕ = M_⊕W_[k,k],

M_⊕² = M_⊕ (I_k ⊗ M_⊕),

$M_{\oplus} M_{n} δ_{k}^{k} = {1_{k}}^{T} \otimes (M_{n} δ_{k}^{k})$ ,

$M_{\oplus} (I_{k} \otimes M_{n}) {PR}_{k} = {1_{k}}^{T} \otimes (M_{n} δ_{k}^{k})$ ,

M_⊕M_nM_⊕ (I_k ⊗ M_B²) (I_k
² ⊗ M_B²) PR_k² $= {1_{k}}^{T} \otimes (M_{n} δ_{k}^{k}),$

$M_{\oplus} M_{n} (I_{k} \otimes {M_{B}}^{2}) {PR}_{k} = {1_{k}}^{T} \otimes (M_{n} δ_{k}^{k})$ ,

M_c (I_k ⊗ M_B²) PR_k = I_k,

M_nM_B = M_B²,

M_BM_c = M_d (M_B ⊗ M_B),

M_BM_d = M_c (M_B ⊗ M_B),

$M_{c} (I_{k} \otimes M_{B}) {PR}_{k} = {1_{k}}^{T} \otimes δ_{k}^{k},$

M_B = M_B³,

M_⊕ (M_B ⊗ M_B) PR_k = M_B,

$M_{n} δ_{k}^{k} = M_{B} M_{n} δ_{k}^{k} .$

Proof. Recalling the Proposition 7.5 in Gattaneo [1], the corresponding algebraic form can be obtained. We only prove (6), other items can be proved similarly. Expressing ¬x ⊕ ∼∼ x = 1 into algebraic form, we have $M_{\oplus} (M_{n} x {M_{B}}^{2} x) = M_{n} δ_{k}^{k},$ the left hand side is $\begin{matrix} M_{\oplus} (M_{n} x {M_{B}}^{2} x) & = M_{\oplus} M_{n} (I_{k} \otimes {M_{B}}^{2}) x^{2} \\ = M_{\oplus} M_{n} (I_{k} \otimes {M_{B}}^{2}) {PR}_{k} x, \end{matrix}$ Since x ∈ Δ_k are arbitrary, (6) follows.□

Connectives ∨ and ∧ are the algebraic realization of logical disjunction and conjunction of a distributive lattice. Then the order is induced as follows.

Definition 3.6. [4] An order “≤” is defined as x ≤ y if and only if x ∨ y = y (equivalently x ∧ y = x), as the notation x < y usually means that x ≤ y and x ≠ y.

BZMV^dM-algebra has the following relationship.

Theorem 3.7. [1] Let $A = 〈 A, \oplus, \neg, \sim, 0 〉$ be a BZMV^dM-algebra, the substructure $A_{MV} = 〈 A, \oplus, \neg, 0 〉$ is an MV-algebra. The substructure $A_{BZ} = 〈 A, \land, \lor, \neg,$ ∼, 0〉 is a distributive BZ-lattice.

The unary operation ¬ : A ↦ A is a Kleene (or Zadeh) orthocomplementation (negation). It satisfies the properties: (K1) ¬ (¬ a) = a, (K2) ¬ (a ∨ b) = ¬ a ∧ ¬ b, (K3) a ∧ ¬ a ≤ b ∨ ¬ b.

The unary operation ∼ : A ↦ A is a Brouwer orthocomplement (negation). It satisfies the properties: (B1) a ∧ ∼∼ a = a, (a ≤ ∼∼ a), (B2) ∼ (a ∨ b) = ∼ a ∧ ∼ b, (B3) a ∧ ∼ a = 0.

Now we introduce some equivalent conditions of BZMV^dM-algebra. According to Proposition 7.6 in the Appendix, the following result is straightforward verifiable.

Proposition 3.8. Assume 〈A, ⊕, ¬, ∼, 0〉 is a BZMV^dM-algebra with |A| = k< ∞, for any x, y, z ∈ Δ_k the following holds:

$M_{c} xy = δ_{k}^{k} \Leftrightarrow M_{c} (I_{k} \otimes M_{B}) W_{[k, k]} xy = y,$

With M_⊕PR_kx = x, then $M_{c} xy = δ_{k}^{k} \Leftrightarrow$ M_c (I_k ⊗ M_n) xy = x,

$M_{B}^{2} = I_{k} \Leftrightarrow M_{\oplus} M_{B} {PR}_{k} x = M_{n} δ_{k}^{k} \Leftrightarrow M_{\oplus} {PR}_{k} x = x,$

$M_{c} M_{\oplus} xyz = δ_{k}^{k} \Leftrightarrow M_{c} xz = δ_{k}^{k}$ ⇔ $M_{c} yz = δ_{k}^{k},$

$M_{c} M_{\oplus} xyz = δ_{k}^{k} \Leftrightarrow M_{c} (I_{k} \otimes M_{⊙}) (I_{k} \otimes M_{B}) (I_{k^{2}} \otimes M_{B}) W_{[k^{2}, k]} xyz = z,$

M_BM_⊕ = M_⊙ (M_B ⊗ M_B).

4 Homomorphism and isomorphism

In this section, we study the homomorphism and isomorphism of the BZMV^dM-algebra by using the matrix expression method. Moreover, the direct verifiable conditions for detecting homomorphism are obtained by logical matrix operation.

4.1 Homomorphism

This subsection gives the necessary and sufficient conditions for algebra homomorphism.

Definition 4.1. [1] Let 〈A, ⊕ ₁, ¬ ₁, ∼ ₁, 0₁〉 and 〈B, ⊕ ₂, ¬₂, ∼ ₂, 0₂〉 be two BZMV^dM-algebras. We say that the function φ : A → B is a homomorphism of A onto B if $\begin{matrix} φ (x \oplus_{1} y) & = φ (x) \oplus_{2} φ (y), x, y \in A . \\ φ (\neg_{1} x) & = \neg_{2} φ (x) . \\ φ (\sim_{1} x) & = \sim_{2} φ (x) . \\ φ (0_{1}) & = 0_{2} . \end{matrix}$

If φ is a homomorphic mapping, we have $φ (1_{1}) = 1_{2}, φ (x ⊙_{1} y) = φ (x) ⊙_{2} φ (y) .$

Assume |A| = p, |B| = q. Express $A = {δ_{p}^{1}, δ_{p}^{2},$ $\dots, δ_{p}^{p}}, B = {δ_{q}^{1}, δ_{q}^{2}, \dots, δ_{q}^{q}} .$ Then φ : A → B can be expressed in a matrix form as $φ (x) = M_{φ} x,$ where $M_{φ} \in L_{q \times p}$ is the structure matrix of φ.

Proposition 4.2. Let 〈A, ⊕ ₁, ¬ ₁, ∼ ₁, 0₁〉 and 〈B, ⊕ ₂, ¬₂, ∼ ₂, 0₂〉 be two BZMV^dM-algebras, |A| = p < ∞, |B| = q< ∞, and φ : A → B. Then φ is a homomorphism, if and only if, $\begin{matrix} M_{φ} M_{\oplus}^{1} = M_{\oplus}^{2} M_{φ} (I_{p} \otimes M_{φ}), \\ M_{φ} M_{n}^{1} = M_{n}^{2} M_{φ}, \\ M_{φ} M_{B}^{1} = M_{B}^{2} M_{φ}, \\ {Col}_{p} (M_{φ}) = δ_{q}^{q} . \end{matrix}$

If φ is a homomorphic mapping, we have $\begin{matrix} M_{φ} {Col}_{p} (M_{n}) = {Col}_{q} (M_{n}), \\ M_{φ} M_{⊙}^{1} = M_{⊙}^{2} M_{φ} (I_{p} \otimes M_{φ}) . \end{matrix}$

Proof. It is sufficient to prove that each formula in the Definition 4.1 is equivalent to the formula in this proposition, so we consider the first formula $φ (x \oplus_{1} y) = φ (x) \oplus_{2} φ (y),$ the matrix form is $\begin{matrix} M_{φ} M_{\oplus}^{1} xy & = M_{\oplus}^{2} M_{φ} x M_{φ} y \\ = M_{\oplus}^{2} M_{φ} (I_{p} \otimes M_{φ}) xy . \end{matrix}$

So, we have $M_{φ} M_{\oplus}^{1} = M_{\oplus}^{2} M_{φ} (I_{p} \otimes M_{φ})$ .□

4.2 Isomorphism

In this subsection, we discuss the properties of isomorphism and give some examples.

Definition 4.3. [1] Let 〈A, ⊕ ₁, ¬ ₁, ∼ ₁, 0₁〉 and 〈B, ⊕ ₂, ¬₂, ∼ ₂, 0₂〉 be two BZMV^dM-algebras. φ : A → B is a homomorphism. If the function φ is one-to-one and onto, then φ is called isomorphism.

Remark 4.4. If φ is an isomorphism, then φ^-1 is also an isomorphism.

When the mapping and operations are expressed as concrete logical matrix operations, the isomorphism satisfies the following conditions.

Proposition 4.5. Assume 〈A, ⊕ ₁, ¬ ₁, ∼ ₁, 0₁〉 and 〈B, ⊕ ₂, ¬ ₂, ∼ ₂, 0₂〉 with |A| = |B| = p be two BZMV^dM-algebras.

(1) φ : A → B is an isomorphism, then

i) The structure matrix $M_{φ} \in L_{p \times p}$ of φ is a permutation matrix, and $M_{φ}^{T} = M_{φ}^{- 1}$ .

ii) The corresponding structure matrices of $M_{\oplus}^{i}, M_{n}^{i},$ $M_{B}^{i}, (i = 1, 2)$ satisfies $M_{\oplus}^{1} = M_{φ}^{T} M_{\oplus}^{2} (M_{φ} \otimes M_{φ}),$ (11) $M_{n}^{1} = M_{φ}^{T} M_{n}^{2} M_{φ},$ (12) $M_{B}^{1} = M_{φ}^{T} M_{B}^{2} M_{φ} .$ (13) (2) A and B is isomorphism if and only if exist φ satisfied M_φ is a permutation matrix and ${Col}_{p} (M_{φ}) = δ_{p}^{p}$ such that Equation (11)–(13) established.

Proof. We just prove Equation (11), the others are similar. According to the Proposition 7.2 and 4.2, $\begin{matrix} M_{\oplus}^{1} & = M_{φ}^{- 1} M_{\oplus}^{2} M_{φ} (I_{p} \otimes M_{φ}) \\ = M_{φ}^{T} (M_{\oplus}^{2} ⋉ M_{φ}) (I_{p} \otimes M_{φ}) \\ = M_{φ}^{T} M_{\oplus}^{2} (M_{φ} \otimes I_{p}) (I_{p} \otimes M_{φ}) \\ = M_{φ}^{T} M_{\oplus}^{2} (M_{φ} \otimes M_{φ}) . \end{matrix}$

□

Here are two examples of determining the isomorphism between finite BZMV^dM-algebras.

Example 4.6. When k=3, the only non-trivial isomorphism, which keep 0 unchanged, is M_φ = δ₃ [2, 1, 3].

Recall Example 3.3, it is easy to see that $M_{\oplus}^{i} (3),$ $M_{n}^{i} (3)$ and $M_{B}^{i} (3) (i = 1, 2)$ satisfy the Equation (11)–(13). Hence, we have isomorphic BZMV^dM-algebra as $A_{1} : = {M_{\oplus}^{1} (3), M_{n}^{1} (3), M_{B}^{1} (3)}$ is isomorphic to $A_{2} : = {M_{\oplus}^{2} (3), M_{n}^{2} (3), M_{B}^{2} (3)}$ .

Example 4.7. Recall Example 3.3. When k=4, there are five non-trivial isomorphisms, which keep 0 unchanged, they are $M_{φ}^{1} = δ_{4} [1, 3, 2, 4], M_{φ}^{2} = δ_{4} [2, 1, 3, 4], M_{φ}^{3} = δ_{4} [2, 3, 1, 4], M_{φ}^{4} = δ_{4} [3, 1, 2, 4], M_{φ}^{5} = δ_{4} [3, 2, 1, 4] .$

We can see that $M_{φ}^{j} = {(M_{φ}^{j})}^{- 1}, (j = 1, 2, 5)$ . If φ_j : A_p → A_q is an isomorphism, then φ_j : A_q → A_p is also an isomorphism. Similarly, $M_{φ}^{3} = {(M_{φ}^{4})}^{- 1}$ , if φ₃ (φ₄) : A_p → A_q is an isomorphism, then φ₄ (φ₃) : A_q → A_p is also an isomorphism.

We set $A_{i} = {M_{\oplus}^{i} (4), M_{n}^{i} (4), M_{B}^{i} (4)}, i = 1, \dots, 9$ , it is easy to verify that the following mappings are isomorphisms: $\begin{matrix} φ_{1} : & A_{1} \to A_{3}; (A_{3} \to A_{1}); A_{2} \to A_{2}; \\ A_{4} \to A_{5}; (A_{5} \to A_{4}); A_{6} \to A_{9}; \\ (A_{9} \to A_{6}); A_{7} \to A_{8}; (A_{8} \to A_{7}); \\ φ_{2} : & A_{1} \to A_{6}; (A_{6} \to A_{1}); A_{2} \to A_{4}; \\ (A_{4} \to A_{2}); A_{3} \to A_{8}; (A_{8} \to A_{3}); \\ A_{5} \to A_{5}; A_{7} \to A_{9}; (A_{9} \to A_{7}); \\ φ_{3} : & A_{1} \to A_{8}; A_{2} \to A_{4}; A_{3} \to A_{6}; \\ A_{4} \to A_{5}; A_{5} \to A_{2}; A_{6} \to A_{7}; \\ A_{7} \to A_{3}; A_{8} \to A_{9}; A_{9} \to A_{1}; \\ φ_{4} : & A_{1} \to A_{9}; A_{2} \to A_{5}; A_{3} \to A_{7}; \\ A_{4} \to A_{2}; A_{5} \to A_{4}; A_{6} \to A_{3}; \\ A_{7} \to A_{6}; A_{8} \to A_{1}; A_{9} \to A_{8}; \\ φ_{5} : & A_{1} \to A_{7}; (A_{7} \to A_{1}); A_{2} \to A_{5}; \\ (A_{5} \to A_{2}); A_{3} \to A_{9}; (A_{9} \to A_{3}); \\ A_{4} \to A_{4}; A_{6} \to A_{8}; (A_{8} \to A_{6}) . \end{matrix}$ In conclusions $A_{1} ≅ A_{3} ≅ A_{6} ≅ A_{7} ≅ A_{8} ≅ A_{9},$ $A_{2} ≅ A_{4} ≅ A_{5} .$ According to isomorphism, this finite BZMV^dM-algebra is divided into two classes.

5 Product of BZMV^dM-algebras

This section mainly discusses the cartesian product of BZMV^dM-algebras, and gives the calculation method of the structure matrix of its operator. First, we give the definition of the product BZMV^dM-algebras.

Definition 5.1. Let L_i = 〈A_i, ⊕ _i, ¬ _i, ∼ _i, 0_i〉, i = 1, 2 be two BZMV^dM-algebras. Their product can be defined as L = 〈A, ⊕, ¬, ∼, 0〉, where A = A₁ × A₂ is the cartesian (or direct) product of A₁ and A₂. Let (x₁, x₂), (y₁, y₂) ∈ A₁ × A₂, define the operator on A as follows:

⊕ : = ⊕ ₁ × ⊕ ₂, $(x_{1}, x_{2}) \oplus (y_{1}, y_{2}) : = (x_{1} \oplus_{1} y_{1}, x_{2} \oplus_{2} y_{2}) .$ (14)

¬ : = ¬ ₁ × ¬ ₂, $\neg (x_{1}, x_{2}) : = (\neg_{1} x_{1}, \neg_{2} x_{2}) .$ (15)

∼ : = ∼ ₁ × ∼ ₂, $\sim (x_{1}, x_{2}) : = (\sim_{1} x_{1}, \sim_{2} x_{2}) .$ (16)

0 : = (0₁, 0₂).

Remark 5.2. According to the Definition 5.1, the product of BZMV^dM-algebras is a BZMV^dM-algebra. The cartesian product of a finite number of algebras can be similarly defined, and the product is also a BZMV^dM-algebra.

Next, consider how to calculate the structure matrix of the operator of product BZMV^dM-algebras.

Theorem 5.3. Let L_i = 〈A_i, ⊕ _i, ¬ _i, ∼ _i, 0_i〉, i = 1, 2 be two BZMV^dM-algebras, |A₁| = p and |A₂| = q. The structure matrices of the operators ⊕_i, ¬ _i, ∼ _i are expressed as $M_{\oplus}^{1} \in L_{p \times p^{2}}, M_{n}^{1}, M_{B}^{1} \in L_{p \times p}$ and $M_{\oplus}^{2} \in L_{q \times q^{2}}, M_{n}^{2}, M_{B}^{2} \in L_{q \times q} .$ Then, the structural matrices of the corresponding operators of product BZMV^dM-algebras are denoted as $M_{\oplus}^{p}, M_{n}^{p}, and M_{B}^{p}$ . It can be calculated as follows: $M_{\oplus}^{p} = (M_{\oplus}^{1} \otimes M_{\oplus}^{2}) (I_{p} \otimes W_{[q, p]}),$ (17) $M_{n}^{p} = M_{n}^{1} (I_{p} \otimes M_{n}^{2}),$ (18) $M_{B}^{p} = M_{B}^{1} (I_{p} \otimes M_{B}^{2}) .$ (19)

Proof. Firstly, the elements x ∈ A₁ are expressed in vector form as $\vec{x} \in Δ_{p}$ , y ∈ A₂ can be expressed as $\vec{y} \in Δ_{q}$ . Similarly, z ∈ A = A₁ × A₂ can be expressed as $\vec{z} \in Δ_{pq}$ . In particular, if $z = (x, y) \in A, (\vec{x}, \vec{y}) = (δ_{p}^{i}, δ_{q}^{j})$ , then $\begin{matrix} \vec{z} = (\vec{x}, \vec{y}) = \vec{x} \vec{y} = δ_{p}^{i} δ_{q}^{j} = δ_{pq}^{(i - 1) q + j}, \end{matrix}$ in which i = 1, ⋯, p ; j = 1, ⋯, q. Under this expression, we have $\vec{0} = ({\vec{0}}_{1}, {\vec{0}}_{2}) = δ_{p}^{p} δ_{q}^{q} = δ_{pq}^{pq} .$

We just prove Equation (17), the others are similar. Let x₁, y₁ ∈ Δ_p, x₂, y₂ ∈ Δ_q. Then we know that $\begin{matrix} (x_{1}, x_{2}) \oplus (y_{1}, y_{2}) = M_{\oplus}^{p} x_{1} x_{2} y_{1} y_{2}, \\ (x_{1} \oplus_{1} y_{1}, x_{2} \oplus_{2} y_{2}) \\ = M_{\oplus}^{1} x_{1} y_{1} M_{\oplus}^{2} x_{2} y_{2} \\ = M_{\oplus}^{1} (I_{p^{2}} \otimes M_{\oplus}^{2}) x_{1} y_{1} x_{2} y_{2} \\ = M_{\oplus}^{1} (I_{p^{2}} \otimes M_{\oplus}^{2}) x_{1} W_{[q, p]} x_{2} y_{1} y_{2} \\ = (M_{\oplus}^{1} \otimes I_{q}) (I_{p^{2}} \otimes M_{\oplus}^{2}) (I_{p} \otimes W_{[q, p]}) x_{1} x_{2} y_{1} y_{2} \\ = (M_{\oplus}^{1} \otimes M_{\oplus}^{2}) (I_{p} \otimes W_{[q, p]}) x_{1} x_{2} y_{1} y_{2}, \end{matrix}$ which is equivalent to $M_{\oplus}^{p} = (M_{\oplus}^{1} \otimes M_{\oplus}^{2}) (I_{p} \otimes W_{[q, p]}) .$ □ An example of solving the product of BZMV^dM-algebras is given below.

Example 5.4. Let L_i = 〈A_i, ⊕ _i, ¬ _i, ∼ _i, 0_i〉, i = 1, 2 be two BZMV^dM-algebras. |A₁|=2, |A₂|=3. Recall Example 3.3, let the structure matrices of L₁ be Equation (2)-(4), the structure matrices of L₂ be Equation (5), according to Equation (17)–(19) we can get $\begin{matrix} M_{\oplus}^{p} (6) = δ_{6} [1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, \\ 1, 2, 3, 1, 1, 1, 4, 4, 4, 1, 1, 2, 4, 4, 5, 1, 2, 3, 4, 5, 6], \\ M_{n}^{p} (6) = δ_{6} [6, 5, 4, 3, 2, 1], \\ M_{B}^{p} (6) = δ_{6} [6, 6, 4, 3, 3, 1] . \end{matrix}$

Let the structure matrices of L₂ be Equation (6), then $\begin{matrix} M_{\oplus}^{p} (6) = δ_{6} [2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, \\ 1, 2, 3, 2, 2, 1, 5, 5, 4, 2, 2, 2, 5, 5, 5, 1, 2, 3, 4, 5, 6], \\ M_{n}^{p} (6) = δ_{6} [4, 6, 5, 1, 3, 2], \\ M_{B}^{p} (6) = δ_{6} [6, 6, 5, 3, 3, 2] . \end{matrix}$

According to Definition 3.2, it is easy to check that the product of L₁ and L₂ is also a BZMV^dM-algebra.

6 Conclusions

BZMV^dM-algebra is a general “fuzzy environment” which can be successfully applied to a number of different fuzzy situations. On the basis of the matrix expression, the finite BZMV^dM-algebra and its related properties are transformed into the semi-tensor product operation of concrete logic matrices. Using this expression method, the homomorphism and isomorphism can be easily determined. Isomorphism classification between finite BZMV^dM-algebras is realized. At the same time, the operation law of cartesian product of two algebras can also be calculated. Therefore, this method is very convenient, which avoids complex abstract operations, and the examples also illustrate its efficiency.

Acknowledgments

The authors are very indebted to the editors and the anonymous referees for valuable comments and suggestions, which greatly improve the original manuscript of this paper. This research is supported by the Natural Science Foundation of Shandong Province (No. ZR2020MA053).

Appendix

The relevant properties of the STP and Kronecker product of matrices are supplemented as follows.

Proposition 7.1. [6] Let A, B, C be real matrix with proper dimensions, $a, b \in ℝ,$ then

A⋉ (aB ± bC) = aA⋉B ± bA⋉C, (aA ± bB) ⋉C = aA⋉C ± bB⋉C.

(A⋉B) ⋉C = A⋉ (B⋉C).

Proposition 7.2. [12]

Let $A \in ℝ^{m \times n}, B \in ℝ^{p \times q}, C \in ℝ^{n \times r} and D \in ℝ^{q \times s}$ . Then $(A \otimes B) (C \otimes D) = (AC) \otimes (BD)$

If A and B are reversible, then $\begin{matrix} A ⋉ B = B^{- 1} ⋉ A^{- 1}, \\ A \otimes B = A^{- 1} \otimes B^{- 1} . \end{matrix}$

Proposition 7.3. [6] Suppose $x_{i} \in ℝ^{d_{i}}, i = 1, \dots, n .$ Let $p = \prod_{i = 1}^{j - 1} d_{i}, q = \prod_{i = j + 2}^{n} d_{i},$ then $\begin{matrix} (I_{p} \otimes W_{[d_{j}, d_{j} + 1]} \otimes I_{q}) ⋉_{i = 1}^{n} x_{i} \\ = x_{1} ⋉ x_{2} ⋉ \dots ⋉ x_{j + 1} ⋉ x_{j} ⋉ \dots ⋉ x_{n} . \end{matrix}$

The definition of the structure matrix in the multilinear mapping is as follows.

Definition 7.4. [6] Let W_i (i = 0, 1, 2, ⋯, n) be vector spaces, the mapping $F : \prod_{i = 1}^{n} W_{i} \to W_{0}$ is a multilinear mapping.

If dim (W_i) = k_i, i = 0, 1, ⋯, n, and the base of W_i is ${δ_{k_{i}}^{1}, δ_{k_{i}}^{2}, \dots, δ_{k_{i}}^{k_{i}}}$ , denote $F (\begin{matrix} δ_{k_{1}}^{j_{1}}, δ_{k_{2}}^{j_{2}}, \dots, δ_{k_{n}}^{j_{n}} \end{matrix}) = \sum_{s = 1}^{k_{0}} c_{s}^{j_{1}, j_{2}, \dots, j_{n}} δ_{k_{0}}^{s},$ j_t = 1, ⋯, k_t, t = 1, ⋯, n. Then ${c_{s}^{j_{1}, j_{2}, \dots, j_{n}} | j_{t} = 1, \dots, k_{t}, t = 1, \dots, n; s = 1, \dots, k_{0}}$ is called structural constants of the mapping F. Arranging the structural constants as a matrix $M_{F} = [\begin{matrix} c_{1}^{11 \dots 1} & \dots & c_{1}^{11 \dots k_{n}} & \dots & c_{1}^{k_{1} k_{2} \dots k_{n}} \\ c_{2}^{11 \dots 1} & \dots & c_{2}^{11 \dots k_{n}} & \dots & c_{2}^{k_{1} k_{2} \dots k_{n}} \\ ⋮ \\ c_{k_{0}}^{11 \dots 1} & \dots & c_{k_{0}}^{11 \dots k_{n}} & \dots & c_{k_{0}}^{k_{1} k_{2} \dots k_{n}} \end{matrix}],$ M_F is called structural matrix of F. Generally, the structural matrix M_F of F is called the algebraic expression of F.

According to the definition of BZMV^dM-algebra, the following operation law can be deduced.

Proposition 7.5. [1] If A is a BZMV^dM-algebra, then the following results are true:

x ⊕ y = y ⊕ x,

(x ⊕ y) ⊕ z = x ⊕ (y ⊕ z),

x ⊕ 1 =1,

x ⊕ ¬ x = 1,

¬ (x ⊕ ∼∼ x) ⊕ ∼∼ x = 1,

¬x ⊕ ∼∼ x = 1,

x ∧ ∼∼ x = x,

¬x ∼ x = ∼∼ x,

∼ (x ∧ y) = ∼ x ∨ ∼ y,

∼ (x ∨ y) = ∼ x ∧ ∼ y,

x ∧ ∼ x = 0,

∼x = ∼∼ ∼ x,

∼x ⊕ ∼ x = ∼ x,

¬0 = ∼1.

Proposition 7.6. [1] Let A be a BZMV^dM-algebra, ∀x, y, z ∈ A, then we have

x ∧ y = 0 iff y ≤ ∼ x (x ≤ ∼ y),

with x ⊕ x = x, x ∧ y = 0 iff x ≤ ¬ y,

∼∼ x = x iff ∼ x ⊕ x = 1 iff x ⊕ x = x,

(x ⊕ y) ∧ z = 0 iff x ∧ z = 0 and y ∧ z = 0,

(x ⊕ y) ∧ z = 0 iff z ≤ ∼ x ⊙ ∼ y,

∼ (x ⊕ y) = ∼ x ⊙ ∼ y.

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Matrix expression of finite BZMV dM -algebra

Abstract

Keywords

1 Introduction

2 Preliminary

3 Matrix expression of finite BZMV dM -algebra

4.1 Homomorphism

4.2 Isomorphism

Acknowledgments

Appendix

References

3 Matrix expression of finite BZMV^dM-algebra