A Construction of Mirror 𝑄-Algebras

Abstract

We investigate how to construct mirror 𝑄-algebras of a 𝑄-algebra, and we obtain the necessary conditions for 𝑀(𝑋) to be a 𝑄-algebra.

1. Introduction

Imai and Iséki introduced two classes of abstract algebras: 𝐵𝐶𝐾-algebras and 𝐵𝐶𝐼-algebras [1, 2]. It is known that the class of 𝐵𝐶𝐾-algebras is a proper subclass of the class of 𝐵𝐶𝐼-algebras. We refer the reader for useful textbooks for 𝐵𝐶𝐾/𝐵𝐶𝐼-algebra to [3–5]. Neggers et al. [6] introduced the notion of 𝑄-algebras which is a generalization of 𝐵𝐶𝐾/𝐵𝐶𝐼/𝐵𝐶𝐻-algebras, obtained several properties, and discussed quadratic 𝑄-algebras. Ahn and Kim [7] introduced the notion of 𝑄𝑆-algebras, and Ahn et al. [8] studied positive implicativity in 𝑄-algebras and discussed some relations between 𝑅−(𝐿−) maps and positive implicativity. Neggers and Kim introduced the notion of 𝑑-algebras which is another useful generalization of 𝐵𝐶𝐾-algebras and then investigated several relations between 𝑑-algebras and 𝐵𝐶𝐾-algebras as well as several other relations between 𝑑-algebras and oriented digraphs [9]. After that some further aspects were studied [10–13]. Allen et al. [14] introduced the notion of mirror image of given algebras and obtained some interesting properties: a mirror algebra of a 𝑑-algebra is also a 𝑑-algebra, and a mirror algebra of an implicative 𝐵𝐶𝐾-algebra is a left 𝐿-up algebra.

In this paper we introduce the notion of mirror algebras to 𝑄-algebras, and we investigate how to construct mirror 𝑄-algebras from a 𝑄-algebra; and we also obtain the necessary conditions for 𝑀(𝑋) to be a 𝑄-algebra.

2. 𝑄-Algebras and Related Algebras

A 𝑄-algebra [6] is a nonempty set 𝑋 with a constant 0 and a binary operation “*” satisfying the following axioms: (I)𝑥∗𝑥=0, (II)0∗𝑥=0, (III)(𝑥∗𝑦)∗𝑧=(𝑥∗𝑧)∗𝑦 for all 𝑥,𝑦,𝑧∈𝑋.

For brevity we also call 𝑋 a 𝑄-algebra. In 𝑋 we can define a binary relation “≤” by 𝑥≤𝑦 if and only if 𝑥∗𝑦=0.

Example 2.1 (see [6]). Let 𝑋∶{0,1,2,3} be a set with the following table: ∗012301000020032320003330(2.1) Then (𝑋,∗,0) is a 𝑄-algebra, which is not a 𝐵𝐶𝐾/𝐵𝐶𝐼/𝐵𝐶𝐻-algebra.
Ahn and Kim [7] introduced the notion of 𝑄𝑆-algebras. They showed that the 𝐺-part of an associative 𝑄𝑆-algebra is a group in which every element is an involution. A 𝑄-algebra 𝑋 is said to be a 𝑄𝑆-algebra if it satisfies the following condition: (IV)(𝑥∗𝑦)∗(𝑥∗𝑧)=𝑧∗𝑦, for all 𝑥,𝑦,𝑧∈𝑋.

Proposition 2.2 (see [6]). If (𝑋,∗,0) is a 𝑄-algebra, then (V)(𝑥∗(𝑥∗𝑦))∗𝑦=0, for all 𝑥,𝑦∈𝑋.

It was proved that every 𝐵𝐶𝐻-algebra is a 𝑄-algebra and every 𝑄-algebra satisfying some additional conditions is a 𝐵𝐶𝐼-algebra.

Neggers and Kim [15] introduced the notion of 𝐵-algebras which is related to several classes of algebras of interest such as 𝐵𝐶𝐻/𝐵𝐶𝐼/𝐵𝐶𝐾-algebras and which seems to have rather nice properties without being excessively complicated otherwise. And they demonstrated some interesting connections between 𝐵-algebras and groups.

Example 2.3. Let 𝑋∶={0,1,2,…,𝜔} be a set. Define a binary operation “∗” on 𝑋 by 𝑥∗𝑦∶=⎧⎪⎨⎪⎩0𝜔𝑥,,,𝑥𝑦𝑦<<≤𝑥𝑥𝑦,,,𝑥𝑦≠=00,.(2.2) Then (𝑋,∗,0) is a 𝑄-algebra, but not a 𝐵-algebra, since (3∗𝜔)∗0=0,  3∗(0∗(0∗𝜔))=3.

Example 2.4. Let 𝑋∶={0,1,…,5} be a set with the following table: ∗012345010213452102453321053443450215453102534210(2.3) Then (𝑋,∗,0) is a 𝐵-algebra, but not a 𝑄-algebra, since (5∗3)∗1=1, (5∗1)∗3=0.

Example 2.5. Let 𝑋 be the set of all real numbers except for a negative integer −𝑛. Define a binary operation ∗ on 𝑋 by 𝑥∗𝑦∶=𝑛(𝑥−𝑦)𝑛+𝑦(2.4) for any 𝑥,𝑦∈𝑋. Then (𝑋,∗,0) is both a 𝑄-algebra and 𝐵-algebra.

If we consider several families of abstract algebras including the well-known 𝐵𝐶𝐾-algebras and several larger classes including the class of 𝑑-algebras which is a generalization of 𝐵𝐶𝐾-algebras, then it is usually difficult and often impossible to obtain a complementation operation and the associated “de Morgan’s laws.” In the sense of this point of view it is natural to construct a “mirror image” of a given algebra which when adjoined to the original algebra permits a natural complementation to take place. The class of 𝐵𝐶𝐾-algebras is not closed under this operation but the class of 𝑑-algebras is, thus explaining why it may be better to work with this class rather than the class of 𝐵𝐶𝐾-algebras. Allen et al. [14] introduced the notion of mirror algebras of a given algebra.

Let (𝑋,∗,0) be an algebra. Let 𝑀(𝑋)∶=𝑋×{0,1}, and define a binary operation “∗” on 𝑀(𝑋) as follows:(𝑥,1)∗((𝑥𝑦,,((00𝑥𝑥)),,∗∶01=))(𝑦∗∗,((1𝑦𝑦)(,,𝑦∶01,=))1∶∶)(==𝑥∗((𝑥𝑦(𝑥∗∗∗𝑥𝑦,,𝑦00))),,,0),(𝑥,1)when𝑥∗𝑦=0,when𝑥∗𝑦≠0.(2.5) Then we say that 𝑀(𝑋)∶=(𝑀(𝑋),∗,(0,0)) is a left mirror algebra of the algebra (𝑋,∗,0). Similarly, if we define (𝑥,∗)∗(𝑦,1)∶=(𝑦∗(𝑦∗𝑥),0),(2.6) then 𝑀(𝑋)∶=(𝑀(𝑋),∗,(0,0)) is a right mirror algebra of the algebra (𝑋,∗,0).

It was shown [14] that the mirror algebra of a 𝑑 (resp., 𝑑-𝐵𝐻)-algebra is also a 𝑑 (resp., 𝑑-𝐵𝐻)-algebra, but the mirror algebra of a 𝐵𝐶𝐾-algebra need not be a 𝐵𝐶𝐾-algebra.

3. A Construction of Mirror 𝑄-Algebras

In [14] Allen et al. defined (left, right) mirror algebras of an algebra, but it is not known how to construct mirror algebras of any given algebra. In this paper, we investigate a construction of a mirror algebra in 𝑄-algebras.

Let (𝑋,∗,0) be a 𝑄-algebra, and let 𝑀(𝑋)∶=𝑋×{0,1}. Define a binary operation “⊕” on 𝑀(𝑋) by(M1)(𝑥,0)⊕(𝑦,0)=(𝑥∗𝑦,0), (M2)(𝑥,1)⊕(𝑦,1)=(𝑦∗𝑥,0), (M3)(𝑥,0)⊕(𝑦,1)=(𝛼(𝑥,𝑦),0), (M4)(𝑥,1)⊕(𝑦,0)=(𝛽(𝑥,𝑦),1),

where 𝛼,𝛽∶𝑋×𝑋→𝑋 are mappings.

Consider condition (I). If we let 𝑥=𝑦 in (1) and (2), then (I) holds trivially. Consider condition (II). For any (𝑥,0)∈𝑀(𝑋), we have (𝑥,0)⊕(0,0)=(𝑥∗0,0)=(𝑥,0). For any (𝑥,1)∈𝑀(𝑋), we have (𝑥,1)=(𝑥,1)⊕(0,0)=(𝛽(𝑥,0),1), which shows that the required condition is 𝛽(𝑥,0)=𝑥. Consider condition (III). There are 8 cases to check that condition (III) holds.

Case 1 ((𝑥,0),(𝑦,0),(𝑧,0)). It holds trivially.

Case 2 ((𝑥,0),(𝑦,1),(𝑧,0)). Since ((𝑥,0)⊕(𝑦,1))⊕(𝑧,0)=(𝛼(𝑥,𝑦),0)⊕(𝑧,0)=(𝛼(𝑥,𝑦)∗𝑧,0) and ((𝑥,0)⊕(𝑧,0))⊕(𝑦,1)=(𝑥∗𝑧,0)⊕(𝑦,1)=(𝛼(𝑥∗𝑧,𝑦),0), we obtain the requirement that 𝛼(𝑥,𝑦)∗𝑧=𝛼(𝑥∗𝑧,𝑦).

Case 3 ((𝑥,0),(𝑦,0),(𝑧,1)). It is the same as Case 2.

Case 4 ((𝑥,0),(𝑦,1),(𝑧,1)). Since ((𝑥,0)⊕(𝑦,1))⊕(𝑧,1)=(𝛼(𝑥,𝑦),0)⊕(𝑧,1)=(𝛼(𝛼(𝑥,𝑦),𝑧),0) and ((𝑥,0)⊕(𝑧,1))⊕(𝑦,1)=(𝛼(𝑥,𝑧),0)⊕(𝑦,1)=(𝛼(𝛼(𝑥,𝑧),𝑦),0), we obtain the requirement that 𝛼(𝛼(𝑥,𝑦),𝑧)=𝛼(𝛼(𝑥,𝑧),𝑦).

Case 5 ((𝑥,1),(𝑦,0),(𝑧,0)). Since ((𝑥,1)⊕(𝑦,0))⊕(𝑧,0)=(𝛽(𝑥,𝑦),1)⊕(𝑧,0)=(𝛽(𝛽(𝑥,𝑦),𝑧),0) and ((𝑥,1)⊕(𝑧,0))⊕(𝑦,0)=(𝛽(𝑥,𝑧),1)⊕(𝑦,0)=(𝛽(𝛽(𝑥,𝑧),𝑦),0), we obtain the requirement that 𝛽(𝛽(𝑥,𝑦),𝑧)=𝛽(𝛽(𝑥,𝑧),𝑦).

Case 6 ((𝑥,1),(𝑦,0),(𝑧,1)). Since ((𝑥,1)⊕(𝑦,0))⊕(𝑧,1)=(𝛽(𝑥,𝑦),1)⊕(𝑧,1)=(𝑧∗𝛽(𝑥,𝑦),0) and ((𝑥,1)⊕(𝑧,1))⊕(𝑦,0)=(𝑧∗𝑥,0)⊕(𝑦,0)=((𝑧∗𝑥)∗𝑦,0), we obtain the requirement that 𝑧∗𝛽(𝑥,𝑦)=(𝑧∗𝑥)∗𝑦.

Case 7 ((𝑥,1),(𝑦,1),(𝑧,0)). It is the same as Case 6.

Case 8 ((𝑥,1),(𝑦,1),(𝑧,1)). Since ((𝑥,1)⊕(𝑦,1))⊕(𝑧,1)=(𝛽(𝑥,𝑦),0)⊕(𝑧,1)=(𝛼(𝛽(𝑥,𝑦),𝑧),0) and ((𝑥,1)⊕(𝑧,1))⊕(𝑦,1)=(𝛼(𝛽(𝑥,𝑧),𝑦),0) by exchanging 𝑦 with 𝑧, we obtain the requirement that 𝛼(𝛽(𝑥,𝑦),𝑧)=𝛼(𝛽(𝑥,𝑧),𝑦). If we summarize this discussion, we obtain the following theorem.

Theorem 3.1. Let (X,∗,0) be a Q-algebra, and let M(X)∶=X×{0,1} be a set with a binary operation “⊕” on M(X) with (𝑀1)∼(𝑀4). Then the necessary conditions for (M(X),⊕,(0,0)) to be a Q-algebra are the following: (i)𝛽(𝑥,0)=𝑥, (ii)𝛼(𝛼(𝑥,𝑦),𝑧)=𝛼(𝛼(𝑥,𝑧),𝑦), (iii)𝛼(𝑥,𝑦)∗𝑧=𝛼(𝑥∗𝑧,𝑦), (iv)𝛽(𝛽(𝑥,𝑦),𝑧)=𝛽(𝛽(𝑥,𝑧),𝑦), (v)𝑧∗𝛽(𝑥,𝑦)=(𝑧∗𝑥)∗𝑦, (vi)𝛼(𝛽(𝑥,𝑦),𝑧)=𝛼(𝛽(𝑥,𝑧),𝑦)for any 𝑥,𝑦,𝑧∈𝑋.

Remark 3.2. By condition (𝑀1), if we identify (𝑥,0)≡𝑥 for any 𝑥∈𝑋, then 𝑋 is a subalgebra of 𝑀(𝑋). By applying Theorem 3.1, we obtain many (mirror) 𝑄-algebras: 𝑋⊆𝑀(𝑋)⊆𝑀(𝑀(𝑋))=𝑀²(𝑋)⊆𝑀³(𝑋)⊆𝑀⁴(𝑋)⊆⋯.

Example 3.3. Let 𝑍 be the set of all integers. Then (𝑍,−,0) is a 𝑄-algebra where “−” is the usual subtraction in 𝑍. If we define mappings 𝛼,𝛽∶𝑍×𝑍→𝑍 by 𝛼(𝑥,𝑦)=𝛽(𝑥,𝑦)=𝑥+𝑦 for any 𝑥,𝑦∈𝑍, then the mirror algebra (𝑀(𝑍),⊕,(0,0)) is also a 𝑄-algebra, that is, (𝑥,0)⊕(𝑦,0)=(𝑥−𝑦,0), (𝑥,1)⊕(𝑦,1)=(𝑦−𝑥,0), (𝑥,0)⊕(𝑦,1)=(𝑥+𝑦,0), and (𝑥,1)⊕(𝑦,0)=(𝑥+𝑦,1).

Example 3.4. Let 𝑋∶={0,1} be a set with the following table: ∗01100010(3.1) Then (𝑋,∗,0) is a 𝑄-algebra. Using the same method we obtain its mirror algebra as follows: 𝑀(𝑋)={0,𝛼,𝛽,𝛾} with the following table: ⊕0𝛼𝛽𝛾𝛼00000𝛽𝛼0𝛼0𝛾𝛽𝛽00𝛾𝛽𝛼0(3.2) where 0∶=(0,0),  𝛼∶=(0,1),  𝛽∶=(1,0), and 𝛾∶=(1,1). It is easy to see that (𝑀(𝑋),⊕,0) is a 𝑄-algebra.

Problems
(1) Find necessary conditions for 𝑀(𝑋) to be a 𝑄𝑆-algebra if (𝑋,∗,0) is a 𝑄𝑆-algebra.
(2) Given a homomorphism 𝑓∶𝑋→𝑌 of 𝑄-algebras, construct a homomorphism 𝑓∶𝑀(𝑋)→𝑀(𝑌) of 𝑄-algebras which is an extension of 𝑓.
(3) Given 𝑄-algebras 𝑋,𝑌, are the mirror algebras 𝑀(𝑀(𝑋×𝑌)) and 𝑀(𝑋)×𝑀(𝑌) isomorphic?