On Cubic KU-Ideals of KU-Algebras

Abstract

We introduce the notion of cubic KU-ideals of KU-algebras and several results are presented in this regard. The image, preimage, and cartesian product of cubic KU-ideals of KU-algebras are defined.

1. Introduction

BCK-algebras form an important class of logical algebras introduced by Iséki and were extensively investigated by several researchers. The class of all BCK-algebras is quasivariety. Iséki and Tanaka introduced two classes of abstract algebras, BCK-algebras and BCI-algebras [1–3]. In connection with this problem, Komori [4] introduced a notion of BCC-algebras.

Prabpayak and Leerawat [5] introduced a new algebraic structure which is called KU-algebra. They gave the concept of homomorphisms of KU-algebras and investigated some related properties in [5, 6].

Zadeh [7] introduced the notion of fuzzy sets. At present this concept has been applied to many mathematical branches, such as groups, functional analysis, probability theory, and topology. Mostafa et al. [8] introduced the notion of fuzzy KU-ideals of KU-algebras and then they investigated several basic properties which are related to fuzzy KU-ideals, also see [9]. Abdullah et al. [10, 11] introduced the concept of direct product of intuitionistic fuzzy sets in BCK-algebras.

Jun et al. [12] introduced the notion of cubic subalgebras/ideals in BCK/BCI-algebras, and then they investigated several properties. They discussed the relationship between a cubic subalgebra and a cubic ideal. Also, they provided characterizations of a cubic subalgebra/ideal and considered a method to make a new cubic subalgebra from an old one, also see [13–17].

In this paper, we introduce the notion of cubic KU-ideals of KU-algebras and then we study the homomorphic image and inverse image of cubic KU-ideals.

2. Preliminaries

In this section we will recall some concepts related to KU-algebra and cubic sets.

Definition 1 (see [5]). By a KU-algebra we mean an algebra (𝑋,∗,0) of type (2,0) with a single binary operation ∗ that satisfies the following identities: for any 𝑥,𝑦,𝑧∈𝑋,(ku1)(𝑥∗𝑦)∗[(𝑦∗𝑧)∗(𝑥∗𝑧)]=0, (ku2)𝑥∗0=0, (ku3) 0∗𝑥=𝑥, (ku4) 𝑥∗𝑦=0=𝑦∗𝑥 implies 𝑥=𝑦.

In what follows, let (𝑋,∗,0) denote a KU-algebra unless otherwise specified. For brevity we also call 𝑋 a KU-algebra. In 𝑋 we can define a binary relation ≤ by: 𝑥≤𝑦 if and only if 𝑦∗𝑥=0.

Definition 2 (see [5]). (𝑋,∗,0) is a KU-algebra if and only if it satisfies(ku5)(𝑦∗𝑧)∗(𝑥∗𝑧)≤(𝑥∗𝑦), (ku6)0≤𝑥, (ku7)𝑥≤𝑦,𝑦≤𝑥 implies 𝑥=𝑦,(ku8)𝑥≤𝑦 if and only if 𝑦∗𝑥=0.

Definition 3 (see [8]). In a KU-algebra, the following identities are true:(1)𝑧∗𝑧=0, (2)𝑧∗(𝑥∗𝑧)=0, (3)𝑥≤𝑦 imply 𝑦∗𝑧≤𝑥∗𝑧,(4)𝑧∗(𝑦∗𝑥)=𝑦∗(𝑧∗𝑥), for all 𝑥,𝑦,𝑧∈𝑋,(5)𝑦∗[(𝑦∗𝑥)∗𝑥]=0.

Example 4 (see [8]). Let 𝑋={0,1,2,3,4} in which ∗ is defined by Table 1.

Table 1 

It is easy to see that 𝑋 is a KU-algebra.

Definition 5 (see [6]). A subset 𝑆 of a KU-algebra 𝑋 is called subalgebra of 𝑋 if 𝑥∗𝑦∈𝑆, whenever 𝑥,𝑦∈𝑆.

Definition 6 (see [6]). A nonempty subset 𝐴 of a KU-algebra 𝑋 is called a KU-ideal of 𝑋 if it satisfies the following conditions:(1)0∈𝐴, (2)𝑥∗(𝑦∗𝑧)∈𝐴,𝑦∈𝐴 implies 𝑥∗𝑧∈𝐴, for all 𝑥,𝑦,𝑧∈𝑋.

Example 7. Let 𝑋={0,1,2,3,4,5} in which ∗ is defined by Table 2.

Table 2 

Clearly (𝑋,∗,0) is a KU-algebra. It is easy to show that 𝐴={0,1} and 𝐵={0,1,2,3,4} are KU-ideals of 𝑋.

Definition 8 (see [6]). Let (𝑋,∗,0) and (𝑋^′,∗^′,0^′) be KU-algebras. A homomorphism is a map 𝑓∶𝑋→𝑋^′ satisfying 𝑓(𝑥∗𝑦)=𝑓(𝑥)∗^′𝑓(𝑦), for all 𝑥,𝑦∈𝑋.

Theorem 9 (see [6]). Let 𝑓 be a homomorphism of a KU-algebra 𝑋 into a KU-algebra 𝑋^′, then(1)if 0 is the identity in 𝑋, then 𝑓(0) is the identity in 𝑋^′;(2)if 𝑆 is a KU-subalgebra of 𝑋, then 𝑓(𝑆) is a KU-subalgebra of 𝑋^′;(3)if 𝐼 is a KU-ideal of 𝑋, then 𝑓(𝐼) is a KU-ideal in 𝑋^′;(4)if 𝑆 is a KU-subalgebra of 𝑓(𝑋^′), then 𝑓⁻¹(𝑆) is a KU-subalgebra algebra of 𝑋;(5)if 𝐵 is a KU-ideal in 𝑓(𝑋^′), then 𝑓⁻¹(𝐵) is a KU-ideal in 𝑋;(6)if 𝑓 is a homomorphism from KU-algebra 𝑋 to a KU-algebra 𝑋^′ then 𝑓 is one to one if and only if ker𝑓={0}.

Proposition 10 (see [8]). Suppose 𝑓∶𝑋→𝑋^′ is a homomorphism of KU-algebras, then:(1)𝑓(0)=0^′, (2)if 𝑥≤𝑦 implies 𝑓(𝑥)≤𝑓(𝑦).

Proposition 11 (see [8]). Let (𝑋,∗,0) and (𝑋^′,∗^′,0^′) be KU-algebras and let 𝑓∶𝑋→𝑋^′ be a homomorphism, then ker𝑓 is a KU-ideal of 𝑋.

Now we will recall the concept of interval-valued fuzzy sets.

An interval number is 𝑎=[𝑎⁻,𝑎⁺], where 0⩽𝑎⁻⩽𝑎⁺⩽1. Let 𝐷[0,1] denote the family of all closed subintervals of [0,1], that is, 𝐷[0,1]=𝑎=𝑎⁻,𝑎⁺∶𝑎⁻⩽𝑎⁺,for𝑎⁻,𝑎⁺∈𝐼.(1) We define the operations “≽,” “⪯,” “=,” “rmin,” and “rmax” in case of two elements in 𝐷[0,1]. We consider two elements 𝑎=[𝑎⁻,𝑎⁺] and 𝑏̃=[𝑏⁻,𝑏⁺] in 𝐷[0,1]. Then(1)𝑎≽𝑏̃ if and only if 𝑎⁻≥𝑏⁻ and 𝑎⁺≥𝑏⁺,(2)𝑎⪯𝑏̃ if and only if 𝑎⁻≤𝑏⁻ and 𝑎⁺≤𝑏⁺,(3)𝑎=𝑏̃ if and only if 𝑎⁻=𝑏⁻ and 𝑎⁺=𝑏⁺,(4)rmin{𝑎,𝑏̃}=[min{𝑎⁻,𝑏⁻},min{𝑎⁺,𝑏⁺}], (5)rmax{𝑎,𝑏̃}=[max{𝑎⁻,𝑏⁻},max{𝑎⁺,𝑏⁺}]. It is obvious that (𝐷[0,1],⪯,∨,∧) is a complete lattice with 0=[0,0] as its least element and 1=[1,1] as its greatest element. Let 𝑎_𝑖∈𝐷[0,1] where 𝑖∈Λ. We define rinf_𝑖∈Λ𝑎_𝑖=inf_𝑖∈Λ𝑎⁻_𝑖,inf_𝑖∈Λ𝑎⁺_𝑖,rsup_𝑖∈Λ𝑎_𝑖=sup_𝑖∈Λ𝑎⁻_𝑖,sup_𝑖∈Λ𝑎⁺_𝑖.(2) An interval-valued fuzzy set (briefly, IVF-set) 𝜇_𝐴 on 𝑋 is defined as 𝜇_𝐴=𝑥,𝜇⁻_𝐴(𝑥),𝜇⁺_𝐴(𝑥)∶𝑥∈𝑋,(3) where 𝜇⁻_𝐴(𝑥)⩽𝜇⁺_𝐴(𝑥), for all 𝑥∈𝑋. Then the ordinary fuzzy sets 𝜇⁻_𝐴∶𝑋→[0,1] and 𝜇⁺_𝐴∶𝑋→[0,1] are called a lower fuzzy set and an upper fuzzy set of 𝜇, respectively. Let 𝜇_𝐴(𝑥)=[𝜇⁻_𝐴(𝑥),𝜇⁺_𝐴(𝑥)], then 𝐴=𝑥,𝜇_𝐴(𝑥)∶𝑥∈𝑋,(4) where 𝜇_𝐴∶𝑋→𝐷[0,1].

Jun et al. [12] introduced the concept of cubic sets defined on a nonempty set 𝑋 as objects having the form: Ξ=𝑥,𝜇_Ξ(𝑥),𝜆_Ξ(𝑥)∶𝑥∈𝑋,(5) which is briefly denoted by Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩, where the functions 𝜇_Ξ∶𝑋→𝐷[0,1] and 𝜆_Ξ∶𝑋→[0,1].

Denote 𝒞(𝑋) by family of all cubic sets in 𝑋.

3. Cubic KU-Ideals of KU-Algebras

In this section, we will introduce a new notion called cubic KU-ideal of KU-algebras and study several properties of it.

Definition 12. Let 𝑋 be a KU-algebra. A cubic set Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ in 𝑋 is called cubic KU-subalgebra of 𝑋 if it satisfies the following conditions:(S1)𝜇_Ξ(0)≽𝜇_Ξ(𝑥) and 𝜆_Ξ(0)≤𝜆_Ξ(𝑥),(S2)𝜇_Ξ(𝑥)≽rmin{𝜇_Ξ(𝑥∗𝑦),𝜇_Ξ(𝑦)} and 𝜆_Ξ(𝑥)≤max{𝜆_Ξ(𝑥∗𝑦),𝜆_Ξ(𝑦)}, for all 𝑥,𝑦∈𝑋.

Definition 13. Let 𝑋 be a KU-algebra. A cubic set Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ in 𝑋 is called cubic KU-ideal of 𝑋 if it satisfies the following conditions:(A1)𝜇_Ξ(0)≽𝜇_Ξ(𝑥) and 𝜆_Ξ(0)≤𝜆_Ξ(𝑥),(A2)𝜇_Ξ(𝑥∗𝑧)≽rmin{𝜇_Ξ(𝑥∗(𝑦∗𝑧)),𝜇_Ξ(𝑦)} and 𝜆_Ξ(𝑥∗𝑧)≤max{𝜆_Ξ(𝑥∗(𝑦∗𝑧)),𝜆_Ξ(𝑦)}, for all 𝑥,𝑦,𝑧∈𝑋.

Example 14. Let 𝑋={0,1,2,3,4} in which ∗ is defined by Table 3.

Table 3 

Clearly (𝑋,∗,0) is a KU-algebra. Define a cubic set Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ in 𝑋 as follows: 𝜇_Ξ(𝑥)=[][]{},[]{},𝜆0.6,0.7,if𝑥=0,0.4,0.5,if𝑥∈1,30.1,0.3,if𝑥∈2,4_Ξ(𝑥)={},{}.0.1,if𝑥=0,0.3,if𝑥∈1,30.6,if𝑥∈2,4(6) By routine calculations it can be seen that the cubic set Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ is a cubic KU-ideal of 𝑋.

Lemma 15. Let Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ be a cubic KU-ideal of KU-algebra 𝑋. If the inequality 𝑥∗𝑦≤𝑧 holds in 𝑋, then 𝜇_Ξ(𝑦)≽ rmin {𝜇_Ξ(𝑥),𝜇_Ξ(𝑧)} and 𝜆_Ξ(𝑦)≤max{𝜆_Ξ(𝑥),𝜆_Ξ(𝑧)}.

Proof. Assume that the inequality 𝑥∗𝑦≤𝑧 holds in 𝑋, then 𝑧∗(𝑥∗𝑦)=0 and by (A2)𝜇_Ξ(𝑧∗𝑦)≽rmin{𝜇_Ξ(𝑧∗(𝑥∗𝑦)),𝜇_Ξ(𝑥)}, if we put 𝑧=0 then 𝜇_Ξ0∗𝑦=𝜇_Ξ𝑦≽rmin𝜇_Ξ0∗𝑥∗𝑦,𝜇_Ξ(𝑥𝜇)=rmin_Ξ𝑥∗𝑦,𝜇_Ξ(𝑥(i)) but𝜇_Ξ𝑥∗𝑦≽rmin𝜇_Ξ𝑥∗𝑧∗𝑦,𝜇_Ξ(𝑧𝜇)=rmin_Ξ𝑧∗𝑥∗𝑦,𝜇_Ξ(𝑧𝜇)=rmin_Ξ(0),𝜇(𝑧)=𝜇_Ξ(𝑧).(ii) From (i) and (ii), we get 𝜇_Ξ(𝑦)≽rmin{𝜇_Ξ(𝑥),𝜇_Ξ(𝑧)}. Similarly we can show that 𝜆_Ξ(𝑦)≤max{𝜆_Ξ(𝑥),𝜆_Ξ(𝑧)}. This completes the proof.

Lemma 16. If Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ is a cubic KU-ideal of KU-algebra 𝑋 and if 𝑥≤𝑦 then 𝜇_Ξ(𝑥)≽𝜇_Ξ(𝑦) and 𝜆_Ξ(𝑥)≤𝜆_Ξ(𝑦).

Proof. If 𝑥≤𝑦 then 𝑦∗𝑥=0. This together with 0∗𝑥=𝑥 and 𝜇_Ξ(0)≽𝜇_Ξ(𝑦) also 𝜆_Ξ(0)≤𝜆_Ξ(𝑦), we get 𝜇_Ξ(0∗𝑥)=𝜇_Ξ(𝑥)≽rmin𝜇_Ξ0∗𝑦∗𝑥,𝜇_Ξ𝑦=rmin𝜇_Ξ(0∗0),𝜇_Ξ𝑦=rmin𝜇_Ξ(0),𝜇_Ξ𝑦=𝜇_Ξ𝑦.(7) Also 𝜆_Ξ(0∗𝑥)=𝜆_Ξ(𝑥)≤max𝜆_Ξ0∗𝑦∗𝑥,𝜆_Ξ𝑦=max𝜆_Ξ(0∗0),𝜆_Ξ𝑦=max𝜆_Ξ(0),𝜆_Ξ𝑦=𝜆_Ξ𝑦.(8) This completes the proof.

Let Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ and ℱ=⟨𝜇_ℱ,𝜆_ℱ⟩ be two cubic sets in a KU-algebra 𝑋, then =𝜇Ξ∩ℱ𝑥,rmin_Ξ(𝑥),𝜇_ℱ(𝑥),max𝜆_Ξ(𝑥),𝜆_ℱ(𝑥=𝜇)∶𝑥∈𝑋𝑥,_Ξ(𝑥)∩𝜇_ℱ(𝑥),𝜆_Ξ(𝑥)∪𝜆_ℱ(𝑥)∶𝑥∈𝑋.(9)

Proposition 17. Let {Ξ_𝑖}_𝑖∈𝐼 be a family of cubic KU-ideals of a KU-algebra 𝑋, then ⋂_𝑖∈𝐼Ξ_𝑖 is a cubic KU-ideal of 𝑋.

Proof. Let {Ξ_𝑖}_𝑖∈𝐼 be a family of cubic KU-ideals of a KU-algebra 𝑋, then for any 𝑥,𝑦,𝑧∈𝑋, 𝜇_Ξ𝑖(0)=rinf𝜇_Ξ𝑖(0)≽rinf𝜇_Ξ𝑖(𝑥)=𝜇_Ξ𝑖𝑥),𝜇_Ξ𝑖(()𝜇𝑥∗𝑧=rinf_Ξ𝑖()𝜇𝑥∗𝑧≽rinfrmin_Ξ𝑖𝑥∗𝑦∗𝑧,𝜇_Ξ𝑖𝑦=rminrinf𝜇_Ξ𝑖𝑥∗𝑦∗𝑧,rinf𝜇_Ξ𝑖𝑦=rmin𝜇_Ξ𝑖𝑥∗𝑦∗𝑧,𝜇_Ξ𝑖𝑦.(10) Also 𝜆_Ξ𝑖(0)=sup𝜆_Ξ𝑖(0)≤sup𝜆_Ξ𝑖(𝑥=𝜆_Ξ𝑖)(𝑥),𝜆_Ξ𝑖()𝜆𝑥∗𝑧=sup_Ξ𝑖()𝜆𝑥∗𝑧≤supmax_Ξ𝑖𝑥∗𝑦∗𝑧,𝜆_Ξ𝑖𝑦=maxsup𝜆_Ξ𝑖𝑥∗𝑦∗𝑧,sup𝜆_Ξ𝑖𝑦=max𝜆_Ξ𝑖𝑥∗𝑦∗𝑧,𝜆_Ξ𝑖𝑦.(11) This completes the proof.

For any 𝑡̃∈𝐷[0,1] and 𝑠∈[0,1], let Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ be a cubic set in a KU-algebra 𝑋, then the set 𝑈Ξ;𝑡,̃𝑠=𝑥∈𝑋∶𝜇_Ξ(𝑥)≽𝑡,̃𝜆_Ξ(𝑥)≤𝑠(12) is called the cubic level set of Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩.

Theorem 18. Let Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ be a cubic subset in 𝑋 then Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ is a cubic KU-ideal of 𝑋 if and only if for all 𝑡̃∈𝐷[0,1] and 𝑠∈[0,1], the set 𝑈(Ξ;𝑡,̃𝑠) is either empty or a KU-ideal of 𝑋.

Proof. Assume that Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ is a cubic KU-ideal of 𝑋, let 𝑡̃∈𝐷[0,1] and 𝑠∈[0,1], be such that 𝑈(Ξ;𝑡,̃𝑠)≠∅, and let 𝑥,𝑦∈𝑋 be such that 𝑥∈𝑈(Ξ;𝑡,̃𝑠), then 𝜇_Ξ(𝑥)≽𝑡̃ and 𝜆_Ξ(𝑥)≤𝑠. By (A2) we get 𝜇_Ξ𝑦∗0=𝜇_Ξ(0)≽rmin𝜇_Ξ𝑦∗(𝑥∗0),𝜇_Ξ(𝑥𝜇)=rmin_Ξ𝑦∗0,𝜇_Ξ(𝑥𝜇)=rmin_Ξ(0),𝜇_Ξ(𝑥)=𝑡,̃(13) also 𝜆_Ξ𝑦∗0=𝜆_Ξ(0)≤max𝜆_Ξ𝑦∗(𝑥∗0),𝜆_Ξ(𝑥𝜆)=max_Ξ𝑦∗0,𝜆_Ξ(𝑥𝜆)=max_Ξ(0),𝜆_Ξ(𝑥)=𝑠.(14) Thus, 0∈𝑈(Ξ;𝑡,̃𝑠). Now letting 𝑥∗(𝑦∗𝑧)∈𝑈(Ξ;𝑡,̃𝑠),𝑦∈𝑈(Ξ;𝑡,̃𝑠), it implies that 𝜇_Ξ(𝑥∗𝑧)≽rmin𝜇_Ξ𝑥∗𝑦∗𝑧,𝜇_Ξ𝑦=𝑡,̃(15) also 𝜆_Ξ(𝑥∗𝑧)≤max𝜆_Ξ𝑥∗𝑦∗𝑧,𝜆_Ξ𝑦=𝑠.(16) So that 𝑥∗𝑧∈𝑈(Ξ;𝑡,̃𝑠). Hence, 𝑈(Ξ;𝑡,̃𝑠) is a KU-ideal of 𝑋.
Conversely, suppose that 𝑈(Ξ;𝑡,̃𝑠) is a KU-ideal of 𝑋 and let 𝑥,𝑦,𝑧∈𝑋 be such that 𝜇_Ξ(𝑥∗𝑧)≺rmin𝜇_Ξ𝑥∗𝑦∗𝑧,𝜇_Ξ𝑦,𝜆_Ξ(𝑥∗𝑧)>max𝜆_Ξ𝑥∗𝑦∗𝑧,𝜆_Ξ𝑦,(17) taking 𝛽₁=12𝜇_Ξ(𝑥∗𝑧)+rmin𝜇_Ξ𝑥∗𝑦∗𝑧,𝜇_Ξ𝑦,(18) and taking𝛽₂=12𝜆_Ξ(𝑥∗𝑧)+max𝜆_Ξ𝑥∗𝑦∗𝑧,𝜆_Ξ𝑦,(19) we have 𝛽₁∈𝐷[0,1] and 𝛽₂∈[0,1], and 𝜇_Ξ(𝑥∗𝑧)≺𝛽₁≺rmin𝜇_Ξ𝑥∗𝑦∗𝑧,𝜇_Ξ𝑦,𝜆_Ξ(𝑥∗𝑧)>𝛽₂>max𝜆_Ξ𝑥∗𝑦∗𝑧,𝜆_Ξ𝑦.(20) It follows that 𝑥∗(𝑦∗𝑧)∈𝑈(Ξ;𝛽₁,𝛽₂) and 𝑥∗𝑧∉𝑈(Ξ;𝛽₁,𝛽₂). This is a contradiction and therefore Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ is cubic KU-ideal of 𝑋.

Proposition 19. If Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ is a cubic KU-ideal of KU-algebra 𝑋, then 𝜇_Ξ𝑥∗𝑥∗𝑦≽𝜇_Ξ𝑦,𝜆_Ξ𝑥∗𝑥∗𝑦≤𝜆_Ξ𝑦.(21)

Proof. Taking 𝑧=𝑥∗𝑦 in (A2) and using (ku2) and (A1), we get 𝜇_Ξ𝑥∗𝑥∗𝑦≽rmin𝜇_Ξ𝑥∗𝑦∗𝑥∗𝑦,𝜇_Ξ𝑦=rmin𝜇_Ξ𝑥∗𝑥∗𝑦∗𝑦,𝜇_Ξ𝑦=rmin𝜇_Ξ(𝑥∗(𝑥∗0)),𝜇_Ξ𝑦=rmin𝜇_Ξ(0),𝜇_Ξ𝑦=𝜇_Ξ𝑦,𝜆_Ξ𝑥∗𝑥∗𝑦≤max𝜆_Ξ𝑥∗𝑦∗𝑥∗𝑦,𝜆_Ξ𝑦=max𝜆_Ξ𝑥∗𝑥∗𝑦∗𝑦,𝜆_Ξ𝑦=max𝜆_Ξ(𝑥∗(𝑥∗0)),𝜆_Ξ𝑦=max𝜆_Ξ(0),𝜆_Ξ𝑦=𝜆_Ξ𝑦.(22) Thus, we get 𝜇_Ξ(𝑥∗(𝑥∗𝑦))≽𝜇_Ξ(𝑦) and 𝜆_Ξ(𝑥∗(𝑥∗𝑦))≤𝜆_Ξ(𝑦). This completes the proof.

4. Image and Preimage of Cubic KU-Ideals

In this section we will present some results on images and preimages of cubic KU-ideals in KU-algebras.

Definition 20. Let 𝑓 be a mapping from a set 𝑋 to a set 𝑌. If Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ is a cubic subset of 𝑋, then the cubic subset 𝛽=⟨𝜇_𝛽,𝜆_𝛽⟩ of 𝑌 is defined by 𝑓𝜇_Ξ𝑦=𝜇_𝛽𝑦=rsup_{𝑥∈𝑓−1(𝑦)}𝜇_Ξ(𝑥),if𝑓⁻¹𝑦=(𝑥)𝑓𝜆𝑥∈𝑋,𝑓=𝑦≠∅,0,otherwise,_Ξ𝑦=𝜆_𝛽𝑦=inf_{𝑥∈𝑓−1(𝑦)}𝜆_Ξ(𝑥),if𝑓⁻¹𝑦=𝑥∈𝑋,𝑓(𝑥)=𝑦≠∅,1,otherwise,(23)is said to be the image of Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ under 𝑓.

Similarly if 𝛽=⟨𝜇_𝛽,𝜆_𝛽⟩ is a cubic subset of 𝑌, then the cubic subset Ξ=𝛽∘𝑓 in 𝑋 (i.e., the cubic subset defined by 𝜇_Ξ(𝑥)=𝜇_𝛽(𝑓(𝑥)) and 𝜆_Ξ(𝑥)=𝜆_𝛽(𝑓(𝑥)) for all 𝑥∈𝑋) is called the preimage of 𝛽 under 𝑓.

Theorem 21. An onto homomorphic preimage of cubic KU-ideal is also cubic KU-ideal.

Proof. Let 𝑓∶𝑋→𝑋^′ be an onto homomorphism of KU-algebras, 𝛽=⟨𝜇_𝛽,𝜆_𝛽⟩ a cubic KU-ideal of 𝑋^′, and Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ the preimage of 𝛽 under 𝑓, then 𝜇_Ξ(𝑥)=𝜇_𝛽(𝑓(𝑥)) and 𝜆_Ξ(𝑥)=𝜆_𝛽(𝑓(𝑥)) for all 𝑥∈𝑋. Let 𝑥∈𝑋, then 𝜇_Ξ(0)=𝜇_𝛽𝑓(0)≽𝜇_𝛽𝑓(𝑥)=𝜇_Ξ(𝑥),𝜆_Ξ(0)=𝜆_𝛽𝑓(0)≤𝜆_𝛽𝑓(𝑥)=𝜆_Ξ(𝑥).(24) Now let 𝑥,𝑦,𝑧∈𝑋, then 𝜇_Ξ()=𝜇𝑥∗𝑧_𝛽𝑓(𝑥∗𝑧)=𝜇_𝛽𝑓(𝑥)∗^′𝑓(𝑧)≽rmin𝜇_𝛽𝑓(𝑥)∗^′𝑓𝑦∗^′𝑓(𝑧),𝜇_𝛽𝑓𝑦=rmin𝜇_𝛽𝑓𝑥∗𝑦∗𝑧,𝜇_𝛽𝑓𝑦=rmin𝜇_Ξ𝑥∗𝑦∗𝑧,𝜇_Ξ𝑦,𝜆_Ξ(𝑥∗𝑧)=𝜆_𝛽𝑓(𝑥∗𝑧)=𝜆_𝛽𝑓(𝑥)∗^′𝑓(𝑧)≤max𝜆_𝛽𝑓(𝑥)∗^′𝑓𝑦∗^′𝑓(𝑧),𝜆_𝛽𝑓𝑦=max𝜆_𝛽𝑓𝑥∗𝑦∗𝑧,𝜆_𝛽𝑓𝑦=max𝜆_Ξ𝑥∗𝑦∗𝑧,𝜆_Ξ𝑦.(25) This completes the proof.

Definition 22. A cubic subset Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ of 𝑋 has sup and inf properties if for any subset 𝑇 of 𝑋, there exist 𝑡,𝑠∈𝑇 such that 𝜇_Ξ(𝑡)=rsup_𝑡∈𝑇𝜇_Ξ(𝑡) and 𝜆_Ξ(𝑠)=inf_𝑡∈𝑇𝜆_Ξ(𝑠).

Theorem 23. Let 𝑓∶𝑋→𝑌 be a homomorphism between KU-algebras 𝑋 and 𝑌. For every cubic KU-ideal Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ in 𝑋,𝑓(Ξ) is cubic KU-ideal of 𝑌.

Proof. By definition 𝜇_𝛽(𝑦^′)=𝑓(𝜇_Ξ)(𝑦^′)=rsup_𝑥∈𝑓⁻¹_(𝑦′)𝜇_Ξ(𝑥) and 𝜆_𝛽(𝑦^′)=𝑓(𝜆_Ξ)(𝑦^′)=inf_𝑥∈𝑓⁻¹_(𝑦′)𝜆_Ξ(𝑥) for all 𝑦^′∈𝑌 and rsup∅=[0,0] and inf∅=0. We have to prove that 𝜇_𝛽(𝑥^′∗𝑧^′)≽rmin{𝜇_𝛽(𝑥^′∗(𝑦^′∗𝑧^′)),𝜇_𝛽(𝑦^′)} and 𝜆_𝛽(𝑥^′∗𝑧^′)≤max{𝜆_𝛽(𝑥^′∗(𝑦^′∗𝑧^′)),𝜆_𝛽(𝑦^′)} for all 𝑥^′,𝑦^′,𝑧^′∈𝑌. Let 𝑓∶𝑋→𝑌 be an onto homomorphism of KU-algebras, Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ a cubic KU-ideal of 𝑋 with sup and inf properties, and 𝛽=⟨𝜇_𝛽,𝜆_𝛽⟩ the image of Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ under 𝑓. Since Ξ=⟨𝜇_Ξ,𝜆_Ξ⟩ is cubic KU-ideal of 𝑋, we have 𝜇_Ξ(0)≽𝜇_Ξ(𝑥),𝜆_Ξ(0)≤𝜆_Ξ(𝑥)∀𝑥∈𝑋.(26) Note that 0∈𝑓⁻¹(0^′) where 0,0^′ are the zero of 𝑋 and 𝑌, respectively. Thus, 𝜇_𝛽0^′=rsup_{𝑡∈𝑓−1(0′)}𝜇_Ξ(𝑡)=𝜇_Ξ(0)≽𝜇_Ξ(𝑥)𝜆∀𝑥∈𝑋,_𝛽0^′=inf_{𝑡∈𝑓−1(0′)}𝜆_Ξ(𝑡)=𝜆_Ξ(0)≤𝜆_Ξ(𝑥)∀𝑥∈𝑋,(27) which implies that 𝜇_𝛽(0^′)≽rsup_𝑡∈𝑓⁻¹_(𝑥′)𝜇_Ξ(𝑡)=𝜇_𝛽(𝑥^′) and 𝜆_𝛽(0^′)≤inf_𝑡∈𝑓⁻¹_(𝑥′)𝜆_Ξ(𝑡)=𝜆_𝛽(𝑥^′) for any 𝑥^′∈𝑌. For any 𝑥^′,𝑦^′,𝑧^′∈𝑌, let 𝑥₀∈𝑓⁻¹(𝑥^′),𝑦₀∈𝑓⁻¹(𝑦^′), and 𝑧₀∈𝑓⁻¹(𝑧^′) be such that 𝜇_Ξ𝑥₀∗𝑧₀=rsup_{𝑡∈𝑓−1(𝑥′∗𝑧′)}𝜇_Ξ(𝑡),𝜇_Ξ𝑦₀=rsup_{𝑡∈𝑓−1(𝑦′)}𝜇_Ξ(𝑡),𝜇_Ξ𝑥₀∗𝑦₀∗𝑧₀=𝜇_𝛽𝑓𝑥₀∗𝑦₀∗𝑧₀=𝜇_𝛽𝑥^′∗𝑦^′∗𝑧^′=rsup_{(𝑥0∗(𝑦0∗𝑧0))∈𝑓−1(𝑥′∗(𝑦′∗𝑧′))}𝜇_Ξ𝑥₀∗𝑦₀∗𝑧₀=rsup_{𝑡∈𝑓−1(𝑥′∗(𝑦′∗𝑧′))}𝜇_Ξ(𝑡).(28) Also 𝜆_Ξ𝑥₀∗𝑧₀=inf_{𝑡∈𝑓−1(𝑥′∗𝑧′)}𝜆_Ξ(𝑡),𝜆_Ξ𝑦₀=inf_{𝑡∈𝑓−1(𝑦′)}𝜆_Ξ(𝑡),𝜆_Ξ𝑥₀∗𝑦₀∗𝑧₀=𝜆_𝛽𝑓𝑥₀∗𝑦₀∗𝑧₀=𝜆_𝛽𝑥^′∗𝑦^′∗𝑧^′=inf_{(𝑥0∗(𝑦0∗𝑧0))∈𝑓−1(𝑥′∗(𝑦′∗𝑧′))}𝜆_Ξ𝑥₀∗𝑦₀∗𝑧₀=inf_{𝑡∈𝑓−1(𝑥′∗(𝑦′∗𝑧′))}𝜆_Ξ(𝑡).(29) Then 𝜇_𝛽𝑥^′∗𝑧^′=rsup_{𝑡∈𝑓−1(𝑥′∗𝑧′)}𝜇_Ξ(𝑡)=𝜇_Ξ𝑥₀∗𝑧₀≽rmin𝜇_Ξ𝑥₀∗𝑦₀∗𝑧₀,𝜇_Ξ𝑦₀=rminrsup_{𝑡∈𝑓−1(𝑥′∗(𝑦′∗𝑧′))}𝜇_Ξ(𝑡),rsup_{𝑡∈𝑓−1(𝑦′)}𝜇_Ξ(𝑡)=rmin𝜇_𝛽𝑥^′∗𝑦^′∗𝑧^′,𝜇_𝛽𝑦^′,𝜆_𝛽𝑥^′∗𝑧^′=inf_{𝑡∈𝑓−1(𝑥′∗𝑧′)}𝜆_Ξ(𝑡)=𝜆_Ξ𝑥₀∗𝑧₀≤max𝜆_Ξ𝑥₀∗𝑦₀∗𝑧₀,𝜆_Ξ𝑦₀=maxinf_{𝑡∈𝑓−1(𝑥′∗(𝑦′∗𝑧′))}𝜆_Ξ(𝑡),inf_{𝑡∈𝑓−1(𝑦′)}𝜆_Ξ(𝑡𝜆)=max_𝛽𝑥^′∗𝑦^′∗𝑧^′,𝜆_𝛽𝑦^′.(30) Hence, 𝛽 is cubic KU-ideal of 𝑌.

5. Cartesian Product of Cubic KU-Ideals

In this section we will provide some new definitions on cartesian product of cubic KU-ideals in KU-algebras.

Definition 24. Let Ξ₁=⟨𝜇_Ξ1,𝜆_Ξ1⟩ and Ξ₂=⟨𝜇_Ξ2,𝜆_Ξ2⟩ be two cubic subsets of KU-algebras 𝑋₁ and 𝑋₂, respectively. Then cartesian product of cubic subsets Ξ₁ and Ξ₂ is denoted by Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ and is defined as 𝜇_Ξ1×Ξ2𝑥,𝑦=rmin𝜇_Ξ1(𝑥),𝜇_Ξ2𝑦,𝜆_Ξ1×Ξ2𝑥,𝑦=max𝜆_Ξ1(𝑥),𝜆_Ξ2𝑦,(31) for all (𝑥,𝑦)∈𝑋₁×𝑋₂.

Remark 25. Let 𝑋 and 𝑌 be KU-algebras. We define ∗ on 𝑋×𝑌 by (𝑥,𝑦)∗(𝑢,𝑣)=(𝑥∗𝑢,𝑦∗𝑣) for every (𝑥,𝑦),(𝑢,𝑣) belong to 𝑋×𝑌, then clearly (𝑋×𝑌,∗,(0,0)) is a KU-algebra.

Definition 26. A cubic subset Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ of 𝑋₁×𝑋₂ is called a cubic KU-subalgebra of 𝑋₁×𝑋₂ if(CP1) 𝜇_Ξ1×Ξ2(0,0)≽𝜇_Ξ1×Ξ2(𝑥,𝑦) and 𝜆_Ξ1×Ξ2(0,0)≤𝜆_Ξ1×Ξ2(𝑥,𝑦),(CP2) 𝜇_Ξ1×Ξ2(𝑥₁,𝑦₁)≽rmin{𝜇_Ξ1×Ξ2((𝑥₁,𝑦₁)∗(𝑥₂,𝑦₂)),𝜇_Ξ1×Ξ2(𝑥₂,𝑦₂)}, (CP3) 𝜆_Ξ1×Ξ2(𝑥₁,𝑦₁)≤max{𝜆_Ξ1×Ξ2((𝑥₁,𝑦₁)∗(𝑥₂,𝑦₂)),𝜆_Ξ1×Ξ2(𝑥₂,𝑦₂)}, for all (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝑋₁×𝑋₂.

Definition 27. A cubic subset Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ of 𝑋₁×𝑋₂ is called a cubic KU-ideal of 𝑋₁×𝑋₂ if(CP4) 𝜇_Ξ1×Ξ2(0,0)≽𝜇_Ξ1×Ξ2(𝑥,𝑦) and 𝜆_Ξ1×Ξ2(0,0)≤𝜆_Ξ1×Ξ2(𝑥,𝑦),(CP5) 𝜇_Ξ1×Ξ2((𝑥₁,𝑦₁)∗(𝑥₃,𝑦₃))≽rmin{𝜇_Ξ1×Ξ2((𝑥₁,𝑦₁)∗((𝑥₂,𝑦₂)∗(𝑥₃,𝑦₃))),𝜇_Ξ1×Ξ2(𝑥₂,𝑦₂)}, (CP6) 𝜆_Ξ1×Ξ2((𝑥₁,𝑦₁)∗(𝑥₃,𝑦₃))≤max{𝜆_Ξ1×Ξ2((𝑥₁,𝑦₁)∗((𝑥₂,𝑦₂)∗(𝑥₃,𝑦₃))),𝜆_Ξ1×Ξ2(𝑥₂,𝑦₂)}, for all (𝑥₁,𝑦₁),(𝑥₂,𝑦₂),(𝑥₃,𝑦₃)∈𝑋₁×𝑋₂.

Theorem 28. Let Ξ₁=⟨𝜇_Ξ1,𝜆_Ξ1⟩ and Ξ₂=⟨𝜇_Ξ2,𝜆_Ξ2⟩ be two cubic KU-subalgebras of KU-algebras 𝑋₁ and 𝑋₂, respectively. Then Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ is a cubic KU-subalgebra of KU-algebra 𝑋₁×𝑋₂.

Proof. For any (𝑥,𝑦)∈𝑋₁×𝑋₂, 𝜇_Ξ1×Ξ2(0,0)=rmin𝜇_Ξ1(0),𝜇_Ξ2(0𝜇)≽rmin_Ξ1(𝑥),𝜇_Ξ2𝑦=𝜇_Ξ1×Ξ2,𝜆𝑥,𝑦_Ξ1×Ξ2(0,0)=max𝜆_Ξ1(0),𝜆_Ξ2(0𝜆)≤max_Ξ1(𝑥),𝜆_Ξ2𝑦=𝜆_Ξ1×Ξ2𝑥,𝑦.(32) For any (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝑋₁×𝑋₂. Then 𝜇_Ξ1×Ξ2𝑥₁,𝑦₁=rmin𝜇_Ξ1𝑥₁,𝜇_Ξ2𝑦₁≽rminrmin𝜇_Ξ1𝑥₁∗𝑥₂,𝜇_Ξ1𝑥₂,rmin𝜇_Ξ2𝑦₁∗𝑦₂,𝜇_Ξ2𝑦₂=rminrmin𝜇_Ξ1𝑥₁∗𝑥₂,𝜇_Ξ2𝑦₁∗𝑦₂,rmin𝜇_Ξ1𝑥₂,𝜇_Ξ2𝑦₂=rmin𝜇_Ξ1×Ξ2𝑥₁∗𝑥₂,𝑦₁∗𝑦₂,𝜇_Ξ1×Ξ2𝑥₂,𝑦₂≽rmin𝜇_Ξ1×Ξ2𝑥₁∗𝑦₁𝑥₂∗𝑦₂,𝜇_Ξ1×Ξ2𝑥₂,𝑦₂,𝜆_Ξ1×Ξ2𝑥₁,𝑦₁=max𝜆_Ξ1𝑥₁,𝜆_Ξ2𝑦₁≤maxmax𝜆_Ξ1𝑥₁∗𝑥₂,𝜆_Ξ1𝑥₂,max𝜆_Ξ2𝑦₁∗𝑦₂,𝜆_Ξ2𝑦₂=maxmax𝜆_Ξ1𝑥₁∗𝑥₂,𝜆_Ξ1𝑦₁∗𝑦₂,max𝜆_Ξ2𝑥₂,𝜆_Ξ2𝑦₂=max𝜆_Ξ1×Ξ2𝑥₁∗𝑥₂,𝑦₁∗𝑦₂,𝜆_Ξ1×Ξ2𝑥₂,𝑦₂≤max𝜆_Ξ1×Ξ2𝑥₁∗𝑦₁𝑥₂∗𝑦₂,𝜆_Ξ1×Ξ2𝑥₂,𝑦₂.(33) Hence, for all (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝑋₁×𝑋₂,Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ is a cubic KU-subalgebra of KU-algebra 𝑋₁×𝑋₂.

Theorem 29. Let Ξ₁=⟨𝜇_Ξ1,𝜆_Ξ1⟩ and Ξ₂=⟨𝜇_Ξ2,𝜆_Ξ2⟩ be two cubic KU-ideals of KU-algebras 𝑋₁ and 𝑋₂, respectively. Then Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ is a cubic KU-ideal of KU-algebra 𝑋₁×𝑋₂.

Proof. For any (𝑥,𝑦)∈𝑋₁×𝑋₂, 𝜇_Ξ1×Ξ2(0,0)=rmin𝜇_Ξ1(0),𝜇_Ξ2(0𝜇)≽rmin_Ξ1(𝑥),𝜇_Ξ2𝑦=𝜇_Ξ1×Ξ2,𝜆𝑥,𝑦_Ξ1×Ξ2(0,0)=max𝜆_Ξ1(0),𝜆_Ξ2(0𝜆)≤max_Ξ1(𝑥),𝜆_Ξ2𝑦=𝜆_Ξ1×Ξ2𝑥,𝑦.(34) Now for any (𝑥₁,𝑦₁),(𝑥₂,𝑦₂),(𝑥₃,𝑦₃)∈𝑋₁×𝑋₂, 𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃=𝜇_Ξ1×Ξ2𝑥₁∗𝑥₃,𝑦₁∗𝑦₃=rmin𝜇_Ξ1𝑥₁∗𝑥₃,𝜇_Ξ2𝑦₁∗𝑦₃≽rminrmin𝜇_Ξ1𝑥₁∗𝑥₂∗𝑥₃,𝜇_Ξ1𝑥₂,rmin𝜇_Ξ2𝑦₁∗𝑦₂∗𝑦₃,𝜇_Ξ2𝑦₂=rminrmin𝜇_Ξ1𝑥₁∗𝑥₂∗𝑥₃,𝜇_Ξ2𝑦₁∗𝑦₂∗𝑦₃,rmin𝜇_Ξ1𝑥₂,𝜇_Ξ2𝑦₂=rmin𝜇_Ξ1×Ξ2𝑥₁∗𝑥₂∗𝑥₃,𝑦₁∗𝑦₂∗𝑦₃,𝜇_Ξ1×Ξ2𝑥₂,𝑦₂≽rmin𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃,𝜇_Ξ1×Ξ2𝑥₂,𝑦₂,𝜆_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃=𝜆_Ξ1×Ξ2𝑥₁∗𝑥₃,𝑦₁∗𝑦₃=max𝜆_Ξ1𝑥₁∗𝑥₃,𝜆_Ξ2𝑦₁∗𝑦₃≤maxmax𝜆_Ξ1𝑥₁∗𝑥₂∗𝑥₃,𝜆_Ξ1𝑥₂,min𝜆_Ξ2𝑦₁∗𝑦₂∗𝑦₃,𝜆_Ξ2𝑦₂=maxmax𝜆_Ξ1𝑥₁∗𝑥₂∗𝑥₃,𝜆_Ξ2𝑦₁∗𝑦₂∗𝑦₃,max𝜆_Ξ1𝑥₂,𝜆_Ξ2𝑦₂=max𝜆_Ξ1×Ξ2𝑥₁∗𝑥₂∗𝑥₃,𝑦₁∗𝑦₂∗𝑦₃,𝜆_Ξ1×Ξ2𝑥₂,𝑦₂≤max𝜆_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃,𝜆_Ξ1×Ξ2𝑥₂,𝑦₂.(35) Hence, for all (𝑥₁,𝑦₁),(𝑥₂,𝑦₂),(𝑥₃,𝑦₃)∈𝑋₁×𝑋₂,Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ is a cubic KU-ideal of KU-algebra 𝑋₁×𝑋₂.

Lemma 30. If Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ is a cubic KU-ideal of KU-algebra 𝑋₁×𝑋₂ and if (𝑥₁,𝑦₁)≤(𝑥₂,𝑦₂), we have 𝜇_Ξ1×Ξ2(𝑥₂,𝑦₂)⪯𝜇_Ξ1×Ξ2(𝑥₁,𝑦₁) and 𝜆_Ξ1×Ξ2(𝑥₂,𝑦₂)≥𝜆_Ξ1×Ξ2(𝑥₁,𝑦₁), for all (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝑋₁×𝑋₂.

Proof. Let (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝑋₁×𝑋₂, such that (𝑥₁,𝑦₁)≤(𝑥₂,𝑦₂)⇒(𝑥₂,𝑦₂)∗(𝑥₁,𝑦₁)=(0,0). This together with (0,0)∗(𝑥₁,𝑦₁)=(𝑥₁,𝑦₁) and 𝜇_Ξ1×Ξ2(𝑥₂,𝑦₂)⪯𝜇_Ξ1×Ξ2(0,0) also 𝜆_Ξ1×Ξ2(𝑥₂,𝑦₂)≥𝜆_Ξ1×Ξ2(0,0). Consider 𝜇_Ξ1×Ξ2(0,0)∗𝑥₁,𝑦₁=𝜇_Ξ1×Ξ2𝑥₁,𝑦₁≽rmin𝜇_Ξ1×Ξ2(0,0)∗𝑥₂,𝑦₂∗𝑥₁,𝑦₁,𝜇_Ξ1×Ξ2𝑥₂,𝑦₂=rmin𝜇_Ξ1×Ξ2((0,0)∗(0,0)),𝜇_Ξ1×Ξ2𝑥₂,𝑦₂=rmin𝜇_Ξ1×Ξ2(0,0),𝜇_Ξ1×Ξ2𝑥₂,𝑦₂=𝜇_Ξ1×Ξ2𝑥₂,𝑦₂,𝜆_Ξ1×Ξ2(0,0)∗𝑥₁,𝑦₁=𝜆_Ξ1×Ξ2𝑥₁,𝑦₁≤max𝜆_Ξ1×Ξ2(0,0)∗𝑥₂,𝑦₂∗𝑥₁,𝑦₁,𝜆_Ξ1×Ξ2𝑥₂,𝑦₂=max𝜆_Ξ1×Ξ2((0,0)∗(0,0)),𝜆_Ξ1×Ξ2𝑥₂,𝑦₂=max𝜆_Ξ1×Ξ2(0,0),𝜆_Ξ1×Ξ2𝑥₂,𝑦₂=𝜆_Ξ1×Ξ2𝑥₂,𝑦₂.(36) This shows that 𝜇_Ξ1×Ξ2(𝑥₂,𝑦₂)⪯𝜇_Ξ1×Ξ2(𝑥₁,𝑦₁) and 𝜆_Ξ1×Ξ2(𝑥₂,𝑦₂)≥𝜆_Ξ1×Ξ2(𝑥₁,𝑦₁), for all (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝑋₁×𝑋₂.

Lemma 31. If Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ is a cubic KU-ideal of KU-algebra 𝑋₁×𝑋₂ and if (𝑥₁,𝑦₁)∗(𝑥₂,𝑦₂)≤(𝑥₃,𝑦₃) holds in 𝑋₁×𝑋₂, then we have 𝜇_Ξ1×Ξ2(𝑥₂,𝑦₂)≽ rmin {𝜇_Ξ1×Ξ2(𝑥₁,𝑦₁),𝜇_Ξ1×Ξ2(𝑥₃,𝑦₃)} and 𝜆_Ξ1×Ξ2(𝑥₂,𝑦₂)≤max{𝜆_Ξ1×Ξ2(𝑥₁,𝑦₁),𝜆_Ξ1×Ξ2(𝑥₃,𝑦₃)}, for all (𝑥₁,𝑦₁),(𝑥₂,𝑦₂),(𝑥₃,𝑦₃)∈𝑋₁×𝑋₂.

Proof. Let (𝑥₁,𝑦₁),(𝑥₂,𝑦₂),(𝑥₃,𝑦₃)∈𝑋₁×𝑋₂ and let (𝑥₁,𝑦₁)∗(𝑥₂,𝑦₂)≤(𝑥₃,𝑦₃) holds in 𝑋₁×𝑋₂, then 𝑥₃,𝑦₃∗𝑥₁,𝑦₁∗𝑥₂,𝑦₂=(0,0).(37) Now for (𝑥₃,𝑦₃)=(0,0) and from (CP5)𝜇_Ξ1×Ξ2𝑥₃,𝑦₃∗𝑥₂,𝑦₂≽rmin𝜇_Ξ1×Ξ2𝑥₃,𝑦₃∗𝑥₁,𝑦₁∗𝑥₂,𝑦₂,𝜇_Ξ1×Ξ2𝑥₁,𝑦₁,(38) we have 𝜇_Ξ1×Ξ2(0,0)∗𝑥₂,𝑦₂=𝜇_Ξ1×Ξ2𝑥₂,𝑦₂≽rmin𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂,𝜇_Ξ1×Ξ2𝑥₁,𝑦₁≽rminrmin𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃𝑥₂,𝑦₂,𝜇_Ξ1×Ξ2𝑥₃,𝑦₃,𝜇_Ξ1×Ξ2𝑥₁,𝑦₁=rminrmin𝜇_Ξ1×Ξ2𝑥₃,𝑦₃∗𝑥₁,𝑦₁𝑥₂,𝑦₂,𝜇_Ξ1×Ξ2𝑥₃,𝑦₃,𝜇_Ξ1×Ξ2𝑥₁,𝑦₁=rminrmin𝜇_Ξ1×Ξ2(0,0),𝜇_Ξ1×Ξ2𝑥₃,𝑦₃,𝜇_Ξ1×Ξ2𝑥₁,𝑦₁=rmin𝜇_Ξ1×Ξ2𝑥₃,𝑦₃,𝜇_Ξ1×Ξ2𝑥₁,𝑦₁=rmin𝜇_Ξ1×Ξ2𝑥₁,𝑦₁,𝜇_Ξ1×Ξ2𝑥₃,𝑦₃,(39) and from (CP6)𝜆_Ξ1×Ξ2𝑥₃,𝑦₃∗𝑥₂,𝑦₂≤max𝜆_Ξ1×Ξ2𝑥₃,𝑦₃∗𝑥₁,𝑦₁∗𝑥₂,𝑦₂,𝜆_Ξ1×Ξ2𝑥₁,𝑦₁,(40) we have 𝜆_Ξ1×Ξ2(0,0)∗𝑥₂,𝑦₂=𝜆_Ξ1×Ξ2𝑥₂,𝑦₂≤max𝜆_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂,𝜆_Ξ1×Ξ2𝑥₁,𝑦₁≤maxmax𝜆_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃𝑥₂,𝑦₂,𝜆_Ξ1×Ξ2𝑥₃,𝑦₃,𝜆_Ξ1×Ξ2𝑥₁,𝑦₁=maxmax𝜆_Ξ1×Ξ2𝑥₃,𝑦₃∗𝑥₁,𝑦₁𝑥₂,𝑦₂,𝜆_Ξ1×Ξ2𝑥₃,𝑦₃,𝜆_Ξ1×Ξ2𝑥₁,𝑦₁=maxmax𝜆_Ξ1×Ξ2(0,0),𝜆_Ξ1×Ξ2𝑥₃,𝑦₃,𝜆_Ξ1×Ξ2𝑥₁,𝑦₁=max𝜆_Ξ1×Ξ2𝑥₃,𝑦₃,𝜆_Ξ1×Ξ2𝑥₁,𝑦₁=max𝜆_Ξ1×Ξ2𝑥₁,𝑦₁,𝜆_Ξ1×Ξ2𝑥₃,𝑦₃.(41) This completes the proof.

Definition 32. Let Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ be a cubic subset of KU-algebra 𝑋₁×𝑋₂, and for any 𝑡̃∈𝐷[0,1] and 𝑠∈[0,1], the set𝑈Ξ₁×Ξ₂;̃=𝑡,𝑠𝑥,𝑦∈𝑋₁×𝑋₂∶𝜇_Ξ1×Ξ2≽̃𝜆𝑥,𝑦𝑡,_Ξ1×Ξ2𝑥,𝑦≤𝑠(42) is called the cubic level set of Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩.

Theorem 33. Let Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ be a cubic subset of KU-algebra 𝑋₁×𝑋₂. Then Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ is a cubic KU-subalgebra of KU-algebra 𝑋₁×𝑋₂ if and only if for any 𝑡̃∈𝐷[0,1] and 𝑠∈[0,1], the set 𝑈(Ξ₁×Ξ₂;𝑡,̃𝑠) is either empty or a KU-subalgebra of 𝑋₁×𝑋₂.

Proof. The proof is straightforward.

Theorem 34. Let Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ be a cubic subset of KU-algebra 𝑋₁×𝑋₂. Then Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ is a cubic KU-ideal of KU-algebra 𝑋₁×𝑋₂ if and only if for any 𝑡̃∈𝐷[0,1] and 𝑠∈[0,1], the set 𝑈(Ξ₁×Ξ₂;𝑡,̃𝑠) is either empty or a KU-ideal of 𝑋₁×𝑋₂.

Proof. Let Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ be a cubic KU-ideal of KU-algebra 𝑋₁×𝑋₂. For any 𝑡̃∈𝐷[0,1] and 𝑠∈[0,1], define the sets 𝑈Ξ₁×Ξ₂;̃=𝑡,𝑠𝑥,𝑦∈𝑋₁×𝑋₂∶𝜇_Ξ1×Ξ2≽̃𝜆𝑥,𝑦𝑡,_Ξ1×Ξ2𝑥,𝑦≤𝑠.(43) Since 𝑈(Ξ₁×Ξ₂;𝑡,̃𝑠)≠∅, let (𝑥,𝑦)∈𝑈(Ξ₁×Ξ₂;𝑡,̃𝑠). This implies 𝜇_Ξ1×Ξ2(𝑥,𝑦)≽𝑡̃ and 𝜆_Ξ1×Ξ2(𝑥,𝑦)≤𝑠. So 𝜇_Ξ1×Ξ2(0,0)≽𝜇_Ξ1×Ξ2𝑥,𝑦≽𝑡̃⇒𝜇_Ξ1×Ξ2()≽̃𝜆0,0𝑡,_Ξ1×Ξ2(0,0)≤𝜆_Ξ1×Ξ2𝑥,𝑦≤𝑠⇒𝜆_Ξ1×Ξ2(0,0)≤𝑠.(44) This shows that (0,0)∈𝑈(Ξ₁×Ξ₂;𝑡,̃𝑠). Let (𝑥₁,𝑦₁)∗((𝑥₂,𝑦₂)∗(𝑥₃,𝑦₃))∈𝑈(Ξ₁×Ξ₂;𝑡,̃𝑠) and (𝑥₂,𝑦₂)∈𝑈(Ξ₁×Ξ₂;𝑡,̃𝑠). This implies 𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃≽̃𝜇𝑡,_Ξ1×Ξ2𝑥₂,𝑦₂≽̃𝜆𝑡,_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃𝜆≤𝑠,_Ξ1×Ξ2𝑥₂,𝑦₂≤𝑠.(45) Since 𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃≽rmin𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃,𝜇_Ξ1×Ξ2𝑥₂,𝑦₂̃𝑡̃=̃𝜆≽rmin𝑡,𝑡,_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃≤max𝜆_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃,𝜆_Ξ1×Ξ2𝑥₂,𝑦₂≤max{𝑠,𝑠}=𝑠.(46) This implied that (𝑥₁,𝑦₁)∗(𝑥₃,𝑦₃)∈𝑈(Ξ₁×Ξ₂;𝑡,̃𝑠). Hence, 𝑈(Ξ₁×Ξ₂;𝑡,̃𝑠) is a KU-ideal of 𝑋₁×𝑋₂.
Conversely, suppose 𝑈(Ξ₁×Ξ₂;𝑡,̃𝑠) is a KU-ideal of 𝑋₁×𝑋₂, for any 𝑡̃∈𝐷[0,1] and 𝑠∈[0,1]. Assume (𝑥₁,𝑦₁)∈𝑋₁×𝑋₂, such that 𝜇_Ξ1×Ξ2(0,0)≺𝜇_Ξ1×Ξ2𝑥₁,𝑦₁,𝜆_Ξ1×Ξ2(0,0)>𝜆_Ξ1×Ξ2𝑥₁,𝑦₁.(47) Put 𝑡_∘=12𝜇_Ξ1×Ξ2(0,0)+𝜇_Ξ1×Ξ2𝑥₁,𝑦₁⇒𝜇_Ξ1×Ξ2(0,0)≺𝑡_∘≺𝜇_Ξ1×Ξ2𝑥₁,𝑦₁,𝑠_∘=12𝜆_Ξ1×Ξ2(0,0)+𝜆_Ξ1×Ξ2𝑥₂,𝑦₂⇒𝜆_Ξ1×Ξ2(0,0)>𝑡_∘>𝜆_Ξ1×Ξ2𝑥₁,𝑦₁.(48) This implies (𝑥₁,𝑦₁)∈𝑈(Ξ₁×Ξ₂;𝑡,̃𝑠), but (0,0)∉𝑈(Ξ₁×Ξ₂;𝑡,̃𝑠), which is contradiction. Therefore, 𝜇_Ξ1×Ξ2(0,0)≽𝜇_Ξ1×Ξ2(𝑥,𝑦) and 𝜆_Ξ1×Ξ2(0,0)≤𝜆_Ξ1×Ξ2(𝑥,𝑦) for all (𝑥,𝑦)∈𝑋₁×𝑋₂. Assume (𝑥₁,𝑦₁),(𝑥₂,𝑦₂),(𝑥₃,𝑦₃)∈𝑋₁×𝑋₂, such that 𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃<rmin𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃,𝜇_Ξ1×Ξ2𝑥₂,𝑦₂.(49) Let 𝑡_∘=12𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃+rmin𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃,𝜇_Ξ1×Ξ2𝑥₂,𝑦₂,(50) then 𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃≺𝑡_∘≺rmin𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃,𝜇_Ξ1×Ξ2𝑥₂,𝑦₂.(51) Also 𝜆_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃>max𝜆_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃,𝜆_Ξ1×Ξ2𝑥₂,𝑦₂.(52) Let 𝑠_∘=12𝜆_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃+max𝜆_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃,𝜆_Ξ1×Ξ2𝑥₂,𝑦₂,(53) then 𝜆_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃>𝑠_∘>max𝜆_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃,𝜆_Ξ1×Ξ2𝑥₂,𝑦₂.(54) This shows that 𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃∈𝑈Ξ₁×Ξ₂;̃,𝑥𝑡,𝑠₂,𝑦₂∈𝑈Ξ₁×Ξ₂;𝑡,̃𝑠.(55) But (𝑥₁,𝑦₁)∗(𝑥₃,𝑦₃)∉𝑈(𝜇_Ξ1×Ξ2;𝑡_∘,𝑠_∘), which is a contradiction; therefore, 𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃≽rmin𝜇_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃,𝜇_Ξ1×Ξ2𝑥₂,𝑦₂.(56) Similarly 𝜆_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₃,𝑦₃≤max𝜆_Ξ1×Ξ2𝑥₁,𝑦₁∗𝑥₂,𝑦₂∗𝑥₃,𝑦₃,𝜆_Ξ1×Ξ2𝑥₂,𝑦₂.(57) Hence, Ξ₁×Ξ₂=⟨𝜇_Ξ1×Ξ2,𝜆_Ξ1×Ξ2⟩ is a cubic KU-ideal of KU-algebra 𝑋₁×𝑋₂.