Characterizing idempotent nullnorms on bounded lattices
aa r X i v : . [ m a t h . G M ] J u l Characterizing idempotent nullnorms on boundedlattices ✩ Xinxing Wu a,b,c , Shudi Liang a , G¨ul Deniz C¸ aylı d, ∗ a School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China b Institute for Artificial Intelligence, Southwest Petroleum University, Chengdu, Sichuan 610500, China c Zhuhai College of Jilin University, Zhuhai, Guangdong 519041, China d Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, 61080 Trabzon, Turkey
Abstract
Nullnorms with a zero element being at any point of a bounded lattice are an important generalizationof triangular norms and triangular conorms. This paper obtains an equivalent characterization for theexistence of idempotent nullnorms with the zero element a on any bounded lattice containing two distinctelements incomparable with a . Furthermore, some basic properties for the bounded lattice containingtwo distinct element incomparable with a are presented. Keywords:
Lattice; Nullnorm; Idempotent nullnorm; Zero element.
1. Introduction
Being the generalizations of triangular norms and triangular conorms, which were introduced bySchweizer and Sklar [18], t-operators and nullnorms were introduced by Mas et al. [14] and Calvo et al.[3], respectively. Then, Mas et al. [15] showed that both of them coincide with each other on [0 , , ,
1] are useful in both theory and applications ([1, 8, 9, 10, 13, 15, 16, 17]). Recently, it hasbeen generalized to bounded lattices by Kara¸cal et al. [12], and extensively investigated. In particular,Kara¸cal et al. [12] proved the existence of nullnorms with the zero element a for arbitrary element a ∈ L \{ , } with underlying t-norms and t-conorms on an arbitrary bounded lattice L . C¸ aylı andKara¸cal [7] showed that there exists a unique idempotent nullnorm on an arbitrary distributive boundedlattice and proved that an idempotent nullnorm on a bounded lattice need not always exists. Ertuˇgrul [11]not only obtained two new methods to construct nullnorms on an arbitrary bounded lattice, but alsointroduced and investigated an equivalent relation based on a class of nullnorms. Xie [21] extendedtype-1 proper nullnorms and proper uninorms to fuzzy truth values and introduced the notions of type-2nullnorms and uninorms. C¸ aylı [6] discussed the characterization of idempotent nullnorms on boundedlattices and researched some of their main properties. Moreover, it was presented some constructionapproaches for nullnorms, in particular idempotent nullnorms, on bounded lattices with the indicatedzero element under some additional constraints. In [4], two new methods were introduced to obtainidempotent nullnorms on bounded lattices with a zero element a under the additional assumption thatall elements incomparable with a are comparable with each other. In [5], four methods were proposed toconstruct nullnorms on bounded lattices derived from triangular norms on [ a, and triangular conormson [0 , a ] , where some sufficient and necessary conditions on theirs zero element a are required. As aconsequence of these methods, some new types idempotent nullnorms on bounded lattices were obtained.Sun and Liu [19] characterized a class of nullnorms on bounded lattices and represented them by triangularnorms and triangular conorms. ✩ This work was supported by the National Natural Science Foundation of China (No. 11601449), the Science and Tech-nology Innovation Team of Education Department of Sichuan for Dynamical System and its Applications (No. 18TD0013),the Key Natural Science Foundation of Universities in Guangdong Province (No. 2019KZDXM027), the Youth Scienceand Technology Innovation Team of Southwest Petroleum University for Nonlinear Systems (No. 2017CXTD02), and theNational Nature Science Foundation of China (Key Program) (No. 51534006). ∗ Corresponding author
Email addresses: [email protected] (Xinxing Wu), shudi [email protected] (Shudi Liang), [email protected] (G¨ul Deniz C¸ aylı)
Preprint submitted to Elsevier July 30, 2020 ¸ aylı and Kara¸cal obtained a result (see [7, Theorem 3]) stating that there exists a unique idempotentnullnorm on bounded lattice L with the zero element a provided that the bounded lattice L contains onlyone element incomparable with a . This paper considers when does the idempotent nullnorm on boundedlattice L with the zero element a exist, under the case that the bounded lattice L contains two distinctelement incomparable with a . By using this equivalent characterization, the bounded lattices which donot admit any idempotent nullnorm on themselves with the zero element a are obtained. Meanwhile,some basic properties for the bounded lattice containing two distinct element incomparable with a areobtained.
2. Preliminaries
This section provides some basic concepts based on bounded lattices and some results related to them.A lattice [2] is a partially ordered set ( L, ≤ ) satisfying that each two elements x, y ∈ L have a greatestlower bound, called infimum and denoted as x ∧ y , as well as a smallest upper bound, called supremum and denoted by x ∨ y . A lattice is called bounded if it has a top element and a bottom element, writtenas 1 and 0, respectively. Let ( L, ≤ , ,
1) denote a bounded lattice with top element 1 and bottom element0 throughout this paper. For x, y ∈ L , the symbol x < y means that x ≤ y and x = y . The elements x and y in L are comparable if x ≤ y or y ≤ x , in this case, we use the notation x ∦ y . Otherwise, x and y are called incomparable if x (cid:2) y and y (cid:2) x , in this case, we use the notation x k y . The set of all elementsof L that are incomparable with a is denoted by I a , i.e., I a = { x ∈ L : x k a } . Definition 1. [2] Let ( L, ≤ , ,
1) be a bounded lattice and a, b ∈ L with a ≤ b . The subinterval [ a, b ] isdefined as [ a, b ] = { x ∈ L : a ≤ x ≤ b } . Other subintervals such as [ a, b ), ( a, b ], and ( a, b ) can be defined similarly. Obviously, ([ a, b ] , ≤ ) is abounded lattice with top element b and bottom element a . Definition 2. [12] Let ( L, ≤ , ,
1) be a bounded lattice. A commutative, associative, non-decreasing ineach variable function V : L → L is called a nullnorm on L if there exists an element a ∈ L , which iscalled a zero element for V , such that V ( x,
0) = x for all x ≤ a and V ( x,
1) = x for all x ≥ a .Clearly, V ( x, a ) = a for all x ∈ L . Definition 3. [4] Let ( L, ≤ , ,
1) be a bounded lattice, a ∈ L \{ , } , and V be a nullnorm on L with thezero element a . V is called an idempotent nullnorm on L if V ( x, x ) = x for all x ∈ L .The following basic properties on the idempotent nullnorms are obtained by C¸ aylı [6], showing thatan idempotent nullnorm on L with the zero element a is completely determined by its values on I a × I a . Lemma 1. [6, Propositions 3, 6 and 7]
Let ( L, ≤ , , be a bounded lattice, a ∈ L \{ , } , and V be anidempotent nullnorm on L with the zero element a . Then, the following statements hold: (i) V ( x, y ) = x ∨ y for all ( x, y ) ∈ [0 , a ] ; (ii) V ( x, y ) = x ∧ y for all ( x, y ) ∈ [ a, ; (iii) V ( x, y ) = x ∨ ( y ∧ a ) for all ( x, y ) ∈ [0 , a ] × I a ; (iv) V ( x, y ) = y ∨ ( x ∧ a ) for all ( x, y ) ∈ I a × [0 , a ] ; (v) V ( x, y ) = x ∧ ( y ∨ a ) for all ( x, y ) ∈ [ a, × I a ; (vi) V ( x, y ) = y ∧ ( x ∨ a ) for all ( x, y ) ∈ I a × [ a, ; (vii) V ( x, y ) = a for all ( x, y ) ∈ ([0 , a ] × [ a, ∪ ([ a, × [0 , a ]) ; (viii) V ( x, y ) = ( x ∧ a ) ∨ ( y ∧ a ) or V ( x, y ) = ( x ∨ a ) ∧ ( y ∨ a ) or V ( x, y ) ∈ I a for all x, y ∈ I a .
3. Characterization of idempotent nullnorms
This section studies idempotent nullnorms with the zero element a on a bounded lattice containingtwo distinct elements incomparable with a and obtains an equivalent characterization for its existence.Moreover, some basic properties for the bounded lattice containing two distinct element incomparablewith a are obtained. 2 emma 2. Let ( L, ≤ , , be a bounded lattice, a ∈ L \{ , } , and V be an idempotent nullnorm on L with the zero element a . For any x, y ∈ I a , if ( x ∧ a ) ∨ ( y ∧ a ) < a and ( x ∨ a ) ∧ ( y ∨ a ) > a , then V ( x, y ) ∈ I a .Proof. For any x, y ∈ I a , consider the following two cases:Case 1. V ( x, y ) = ( x ∧ a ) ∨ ( y ∧ a ) < a . Applying Lemma 1, it follows that V (1 , V ( x, y )) = a and V ( V (1 , x ) , y ) = V ( x ∨ a, y ) = ( x ∨ a ) ∧ ( y ∨ a ) > a , which contracts with the associativity of V ;Case 2. V ( x, y ) = ( x ∨ a ) ∧ ( y ∨ a ) > a . Applying Lemma 1, it follows that V (0 , V ( x, y )) = a and V ( V (0 , x ) , y ) = V ( x ∧ a, y ) = ( x ∧ a ) ∨ ( y ∧ a ) < a , which contracts with the associativity of V .This, together with Lemma 1–(viii), implies that V ( x, y ) ∈ I a . Theorem 1.
Let ( L, ≤ , , be a bounded lattice, a ∈ L \{ , } , and I a = { p, q } with p = q . Then,there exists an idempotent nullnorm on L with the zero element a if, and only if, one of the followingstatements holds: (i) ( p ∧ a ) ∨ ( q ∧ a ) = a ; (ii) ( p ∨ a ) ∧ ( q ∨ a ) = a ; (iii) p ∨ a ≤ q ∨ a and q ∧ a ≤ p ∧ a ; (iv) p ∧ a ≤ q ∧ a and q ∨ a ≤ p ∨ a .Proof. Necessity. Let V be an idempotent nullnorm on L with the zero element a and suppose on the contrary thatnone of statements (i)–(iv) holds. This implies that the following hold:(i ′ ) ( p ∧ a ) ∨ ( q ∧ a ) < a ;(ii ′ ) ( p ∨ a ) ∧ ( q ∨ a ) > a ;(iii ′ ) p ∨ a (cid:2) q ∨ a or q ∧ a (cid:2) p ∧ a ;(iv ′ ) p ∧ a (cid:2) q ∧ a or q ∨ a (cid:2) p ∨ a .Applying Lemma 2, (i ′ ) and (ii ′ ), it follows that V ( p, q ) = V ( q, p ) ∈ I a . Noting that I a = { p, q } , it sufficesto consider the following two cases:(1) If V ( p, q ) = V ( q, p ) = p , then V (0 , V ( p, q )) = V (0 , p ) = p ∧ a, V ( V (0 , p ) , q ) = V ( p ∧ a, q ) = ( p ∧ a ) ∨ ( q ∧ a ) , and V (1 , V ( p, q )) = V (1 , p ) = p ∨ a, V ( V (1 , p ) , q ) = V ( p ∨ a, q ) = ( p ∨ a ) ∧ ( q ∨ a ) . This, together with the associativity of V , implies that p ∧ a = ( p ∧ a ) ∨ ( q ∧ a ) , and p ∨ a = ( p ∨ a ) ∧ ( q ∨ a ) , and thus q ∧ a ≤ p ∧ a and p ∨ a ≤ q ∨ a . This contracts with (iii ′ ).(2) If V ( p, q ) = V ( q, p ) = q , similarly to the above proof, it can be verified that p ∧ a ≤ q ∧ a and q ∨ a ≤ p ∨ a , which contracts with (iv ′ ). Sufficiency. p ∧ a ) ∨ ( q ∧ a ) = a . From [6, Theorem 4], it follows that the binary operation V : L → L definedby V ( x, y ) = x ∨ y, ( x, y ) ∈ [0 , a ] ,x ∧ y, ( x, y ) ∈ [ a, ,a, ( x, y ) ∈ ([0 , a ] × [ a, ∪ ([ a, × [0 , a ]) ,x ∨ ( y ∧ a ) , ( x, y ) ∈ [0 , a ] × I a ,y ∨ ( x ∧ a ) , ( x, y ) ∈ I a × [0 , a ] ,x ∧ ( y ∨ a ) , ( x, y ) ∈ [ a, × I a ,y ∧ ( x ∨ a ) , ( x, y ) ∈ I a × [ a, ,x, x, y ∈ I a and x = y, ( x ∨ a ) ∧ ( y ∨ a ) , ( x, y ) ∈ { ( p, q ) , ( q, p ) } , is an idempotent nullnorm on L with the zero element a .(ii) ( p ∨ a ) ∧ ( q ∨ a ) = a . From [6, Theorem 5], it follows that the binary operation V : L → L definedby V ( x, y ) = x ∨ y, ( x, y ) ∈ [0 , a ] ,x ∧ y, ( x, y ) ∈ [ a, ,a, ( x, y ) ∈ ([0 , a ] × [ a, ∪ ([ a, × [0 , a ]) ,x ∨ ( y ∧ a ) , ( x, y ) ∈ [0 , a ] × I a ,y ∨ ( x ∧ a ) , ( x, y ) ∈ I a × [0 , a ] ,x ∧ ( y ∨ a ) , ( x, y ) ∈ [ a, × I a ,y ∧ ( x ∨ a ) , ( x, y ) ∈ I a × [ a, ,x, x, y ∈ I a and x = y, ( x ∧ a ) ∨ ( y ∧ a ) , ( x, y ) ∈ { ( p, q ) , ( q, p ) } , is an idempotent nullnorm on L with the zero element a .(iii) p ∨ a ≤ q ∨ a and q ∧ a ≤ p ∧ a . We claim that the binary operation V : L → L defined by V ( x, y ) = x ∨ y, ( x, y ) ∈ [0 , a ] ,x ∧ y, ( x, y ) ∈ [ a, ,a, ( x, y ) ∈ ([0 , a ] × [ a, ∪ ([ a, × [0 , a ]) ,x ∨ ( y ∧ a ) , ( x, y ) ∈ [0 , a ] × I a ,y ∨ ( x ∧ a ) , ( x, y ) ∈ I a × [0 , a ] ,x ∧ ( y ∨ a ) , ( x, y ) ∈ [ a, × I a ,y ∧ ( x ∨ a ) , ( x, y ) ∈ I a × [ a, ,x, x, y ∈ I a and x = y,p, ( x, y ) ∈ { ( p, q ) , ( q, p ) } , (3.1)is an idempotent nullnorm on L with the zero element a .First, it can be verified that V is a commutative binary operation with the zero element a . Itremains to check the monotonicity and associativity of V .iii-1) Monotonicity.
For any x, y ∈ L with x ≤ y , it holds that V ( x, z ) ≤ V ( y, z ) for all z ∈ L .Consider the following cases:1. x ∈ [0 , a ). It is easy to see that V ( x, z ) = x ∨ z, z ∈ [0 , a ) ,a, z ∈ [ a, ,x ∨ ( z ∧ a ) , z ∈ I a . (3.2)1.1. If y ∈ [0 , a ), applying (3.2), it is clear that V ( x, z ) ≤ V ( y, z ) holds for all z ∈ L .4.2. If y ∈ [ a, V ( y, z ) = a, z ∈ [0 , a ) ,y ∧ z, z ∈ [ a, ,y ∧ ( z ∨ a ) , z ∈ I a , (3.3) ≥ x ∨ z, z ∈ [0 , a ) ,a, z ∈ [ a, ,a, z ∈ I a , ≥ V ( x, z ) . y = p , it follows from x ≤ y = p , 0 ≤ x < a , and q ∧ a ≤ p ∧ a that p ∧ a ≥ x ∧ a = x , x ∨ ( p ∧ a ) ≤ p ∨ ( p ∧ a ) ≤ p , and x ∨ ( q ∧ a ) ≤ p ∨ ( q ∧ a ) ≤ p ∨ ( p ∧ a ) = p . These,together with (3.1) and (3.2), imply that V ( y, z ) = V ( p, z ) = z ∨ ( p ∧ a ) , z ∈ [0 , a ) ,z ∧ ( p ∨ a ) , z ∈ [ a, ,p, z = p,p, z = q, ≥ z ∨ x, z ∈ [0 , a ) ,a, z ∈ [ a, ,x ∨ ( p ∧ a ) , z = p,x ∨ ( q ∧ a ) , z = q, = V ( x, z ) . y = q , it follows from x ≤ y = q , 0 ≤ x < a , and q ∧ a ≤ p ∧ a that q ∧ a ≥ x ∧ a = x , x ∨ ( p ∧ a ) ≤ ( q ∧ a ) ∨ ( p ∧ a ) = p ∧ a ≤ p , and x ∨ ( q ∧ a ) ≤ q ∨ ( q ∧ a ) = q . These,together with (3.1) and (3.2), imply that V ( y, z ) = V ( q, z ) = z ∨ ( q ∧ a ) , z ∈ [0 , a ) ,z ∧ ( q ∨ a ) , z ∈ [ a, ,q, z = q,p, z = p, ≥ z ∨ x, z ∈ [0 , a ) ,a, z ∈ [ a, ,x ∨ ( q ∧ a ) , z = q,x ∨ ( p ∧ a ) , z = p, = V ( x, z ) . x ∈ [ a, x ≤ y , it is clear that y ∈ [ a, V ( x, z ) ≤ V ( y, z ) holds for all z ∈ L .3. x ∈ I a . From x ≤ y , it follows that y ∈ [ a, ∪ I a . Applying (3.1), it can be verified that V ( x, z ) = z ∨ ( x ∧ a ) , z ∈ [0 , a ) ,z ∧ ( x ∨ a ) , z ∈ [ a, ,x, z = x,p, z ∈ I a \{ x } . (3.4)3.1. If y ∈ I a , consider the following cases:3.1.1. x = y . It is clear that V ( x, z ) ≤ V ( y, z ) holds for all z ∈ L .3.1.2. p = x < y = q . Applying (3.4) yields that V ( x, z ) = V ( p, z ) = z ∨ ( p ∧ a ) , z ∈ [0 , a ) ,z ∧ ( p ∨ a ) , z ∈ [ a, ,p, z = p,p, z = q, V ( y, z ) = V ( q, z ) = z ∨ ( q ∧ a ) , z ∈ [0 , a ) ,z ∧ ( q ∨ a ) , z ∈ [ a, ,p, z = p,q, z = q, implying that V ( x, z ) ≤ V ( y, z ) holds for all z ∈ L .3.1.3. q = x < y = p . Applying (3.4) yields that V ( x, z ) = V ( q, z ) = z ∨ ( q ∧ a ) , z ∈ [0 , a ) ,z ∧ ( q ∨ a ) , z ∈ [ a, ,p, z = p,q, z = q. and V ( y, z ) = V ( p, z ) = z ∨ ( p ∧ a ) , z ∈ [0 , a ) ,z ∧ ( p ∨ a ) , z ∈ [ a, ,p, z = p,p, z = q, implying that V ( x, z ) ≤ V ( y, z ) holds for all z ∈ L .3.2. If y ∈ [ a, x = p . From p = x ≤ y and p ∨ a ≤ q ∨ a , it follows that y ∧ ( p ∨ a ) ≥ x ∧ ( p ∨ a ) = p ∧ ( p ∨ a ) = p and y ∧ ( q ∨ a ) ≥ x ∧ ( q ∨ a ) ≥ x ∧ ( p ∨ a ) = p ∧ ( p ∨ a ) = p . These,together with (3.4) and (3.3), imply that V ( x, z ) = V ( p, z ) = z ∨ ( p ∧ a ) , z ∈ [0 , a ) ,z ∧ ( p ∨ a ) , z ∈ [ a, ,p, z = p,p, z = q, ≤ a, z ∈ [0 , a ) ,z ∧ ( y ∨ a ) , z ∈ [ a, ,y ∧ ( p ∨ a ) , z = p,y ∧ ( q ∨ a ) , z = q, = V ( y, z ) . x = q . From q = x ≤ y , a ≤ y ≤
1, and p ∨ a ≤ q ∨ a , it follows that y ∧ ( p ∨ a ) =( y ∨ a ) ∧ ( p ∨ a ) ≥ ( q ∨ a ) ∧ ( p ∨ a ) = p ∨ a ≥ p and y ∧ ( q ∨ a ) ≥ q ∧ ( q ∨ a ) = q .These, together with (3.4) and (3.3), imply that V ( x, z ) = V ( q, z ) = z ∨ ( q ∧ a ) , z ∈ [0 , a ) ,z ∧ ( q ∨ a ) , z ∈ [ a, ,q, z = q,p, z = p, ≤ a, z ∈ [0 , a ) ,z ∧ ( y ∨ a ) , z ∈ [ a, ,y ∧ ( q ∨ a ) , z = q,y ∧ ( p ∨ a ) , z = p, = V ( y, z ) . iii-2) Associativity.
For any x, y, z ∈ L , it holds that V ( x, V ( y, z )) = V ( V ( x, y ) , z ). Consider thefollowing cases:1. x ∈ [0 , a ].1.1. y ∈ [0 , a ].1.1.1. If z ∈ [0 , a ], this holds trivially.6.1.2. If z ∈ [ a, V ( x, V ( y, z )) = V ( x, a ) = a = V ( x ∨ y, z ) = V ( V ( x, y ) , z ).1.1.3. If z ∈ I a , then V ( x, V ( y, z )) = V ( x, y ∨ ( z ∧ a )) = x ∨ ( y ∨ ( z ∧ a )) = ( x ∨ y ) ∨ ( z ∧ a ) = V ( x, y ) ∨ ( z ∧ a ) = V ( V ( x, y ) , z ).1.2. y ∈ [ a, z ∈ [0 , a ], then V ( x, V ( y, z )) = V ( x, a ) = a = V ( a, z ) = V ( V ( x, y ) , z ).1.2.2. If z ∈ [ a, V ( x, V ( y, z )) = V ( x, y ∧ z ) = a = V ( a, z ) = V ( V ( x, y ) , z ).1.2.3. If z ∈ I a , then V ( x, V ( y, z )) = V ( x, y ∧ ( z ∨ a )) = a = V ( a, z ) = V ( V ( x, y ) , z ).1.3. y ∈ I a .1.3.1. If z ∈ [0 , a ], then V ( x, V ( y, z )) = V ( x, ( y ∧ a ) ∨ z ) = x ∨ ( y ∧ a ) ∨ z = V ( x, y ) ∨ z = V ( V ( x, y ) , z ).1.3.2. If z ∈ [ a, V ( x, V ( y, z )) = V ( x, ( y ∨ a ) ∧ z ) = a = V ( x ∨ ( y ∧ a ) , z ) = V ( V ( x, y ) , z ).1.3.3. If z ∈ I a , consider the following cases:1.3.3.1. y = z . It can be verified that V ( x, V ( y, z )) = V ( x, y ) = x ∨ ( y ∧ a ) , and V ( V ( x, y ) , z ) = V ( x ∨ ( y ∧ a ) , y ) = x ∨ ( y ∧ a ) ∨ ( y ∧ a ) = x ∨ ( y ∧ a ) , implying that V ( x, V ( y, z )) = V ( V ( x, y ) , z ).1.3.3.2. y = z . Applying q ∧ a ≤ p ∧ a , it can be verified that V ( x, V ( y, z )) = V ( x, p ) = x ∨ ( p ∧ a ) , and V ( V ( x, y ) , z ) = V ( x ∨ ( y ∧ a ) , z ) = x ∨ ( y ∧ a ) ∨ ( z ∧ a ) = x ∨ ( p ∧ a ) , implying that V ( x, V ( y, z )) = V ( V ( x, y ) , z ).2. x ∈ [ a, y ∈ [0 , a ].2.1.1. If z ∈ [0 , a ], then V ( x, V ( y, z )) = V ( V ( z, y ) , x ) (commutativity of V )= V ( z, V ( y, x )) (by 1.1.2)= V ( V ( x, y ) , z ) (commutativity of V ) . z ∈ [ a, V ( x, V ( y, z )) = V ( x, a ) = a = V ( a, z ) = V ( V ( x, y ) , z ).2.1.3. If z ∈ I a , then V ( x, V ( y, z )) = V ( x, y ∨ ( z ∧ a )) = a = V ( a, z ) = V ( V ( x, y ) , z ).2.2. y ∈ [ a, z ∈ [0 , a ), then V ( x, V ( y, z )) = V ( V ( z, y ) , x ) (commutativity of V )= V ( z, V ( y, x )) (by 1.2.2)= V ( V ( x, y ) , z ) (commutativity of V ) . z ∈ [ a, z ∈ I a , then V ( x, V ( y, z )) = V ( x, y ∧ ( z ∨ a )) = x ∧ y ∧ ( z ∨ a ) = V ( x, y ) ∧ ( z ∨ a ) = V ( V ( x, y ) , z ).2.3. y ∈ I a .2.3.1. If z ∈ [0 , a ], then V ( x, V ( y, z )) = V ( V ( z, y ) , x ) (commutativity of V )= V ( z, V ( y, x )) (by 1.3.2)= V ( V ( x, y ) , z ) (commutativity of V ) . z ∈ [ a, V ( x, V ( y, z )) = V ( x, ( y ∨ a ) ∧ z ) = x ∧ ( y ∨ a ) ∧ z = V ( x, y ) ∧ z = V ( V ( x, y ) , z ). 7.3.3. If z ∈ I a , consider the following cases:2.3.3.1. y = z . It can be verified that V ( x, V ( y, z )) = V ( x, y ) = x ∧ ( y ∨ a ) , and V ( V ( x, y ) , z ) = V ( x ∧ ( y ∨ a ) , y ) = x ∧ ( y ∨ a ) ∧ ( y ∨ a ) = x ∧ ( y ∨ a ) , implying that V ( x, V ( y, z )) = V ( V ( x, y ) , z ).2.3.3.2. y = z . Applying p ∨ a ≤ q ∨ a , it can be verified that V ( x, V ( y, z )) = V ( x, p ) = x ∧ ( p ∨ a ) , and V ( V ( x, y ) , z ) = V ( x ∧ ( y ∨ a ) , z ) = x ∧ ( y ∨ a ) ∧ ( z ∨ a ) = x ∧ ( p ∨ a ) , implying that V ( x, V ( y, z )) = V ( V ( x, y ) , z ).3. x ∈ I a .3.1. y ∈ [0 , a ].3.1.1. If z ∈ [0 , a ], then V ( x, V ( y, z )) = V ( V ( z, y ) , x ) (commutativity of V )= V ( z, V ( y, x )) (by 1.1.3)= V ( V ( x, y ) , z ) (commutativity of V ) . z ∈ [ a, V ( x, V ( y, z )) = V ( V ( z, y ) , x ) (commutativity of V )= V ( z, V ( y, x )) (by 2.1.3)= V ( V ( x, y ) , z ) (commutativity of V ) . z ∈ I a , then V ( x, V ( y, z )) = V ( x, y ∨ ( z ∧ a )) = ( x ∧ a ) ∨ y ∨ ( z ∧ a ) = V ( x, y ) ∨ ( z ∧ a ) = V ( V ( x, y ) , z ).3.2. y ∈ [ a, z ∈ [0 , a ], then V ( x, V ( y, z )) = V ( V ( z, y ) , x ) (commutativity of V )= V ( z, V ( y, x )) (by 1.2.3)= V ( V ( x, y ) , z ) (commutativity of V ) . z ∈ [ a, V ( x, V ( y, z )) = V ( V ( z, y ) , x ) (commutativity of V )= V ( z, V ( y, x )) (by 2.2.3)= V ( V ( x, y ) , z ) (commutativity of V ) . z ∈ I a , then V ( x, V ( y, z )) = V ( x, y ∧ ( z ∨ a )) = ( x ∨ a ) ∧ y ∧ ( z ∨ a ) = V ( x, y ) ∧ ( z ∨ a ) = V ( V ( x, y ) , z ).3.3. y ∈ I a .3.3.1. If z ∈ [0 , a ], then V ( x, V ( y, z )) = V ( V ( z, y ) , x ) (commutativity of V )= V ( z, V ( y, x )) (by 1.3.3)= V ( V ( x, y ) , z ) (commutativity of V ) . z ∈ [ a, V ( x, V ( y, z )) = V ( V ( z, y ) , x ) (commutativity of V )= V ( z, V ( y, x )) (by 2.3.3)= V ( V ( x, y ) , z ) (commutativity of V ) . z ∈ I a , it is not difficult to check that V ( x, V ( y, z )) = ( q, x = y = z = q,p, otherwise , and V ( V ( x, y ) , z ) = ( q, x = y = z = q,p, otherwise . (iv) p ∧ a ≤ q ∧ a and q ∨ a ≤ p ∨ a . Similarly to the proof of (iii), it can be verified that the binaryoperation V : L → L defined by V ( x, y ) = x ∨ y, ( x, y ) ∈ [0 , a ] ,x ∧ y, ( x, y ) ∈ [ a, ,a, ( x, y ) ∈ ([0 , a ] × [ a, ∪ ([ a, × [0 , a ]) ,x ∨ ( y ∧ a ) , ( x, y ) ∈ [0 , a ] × I a ,y ∨ ( x ∧ a ) , ( x, y ) ∈ I a × [0 , a ] ,x ∧ ( y ∨ a ) , ( x, y ) ∈ [ a, × I a ,y ∧ ( x ∨ a ) , ( x, y ) ∈ I a × [ a, ,x, x, y ∈ I a and x = y,q, ( x, y ) ∈ { ( p, q ) , ( q, p ) } , (3.5)is an idempotent nullnorm on L with the zero element a . Corollary 1.
If a bounded lattice L contains a sublattice which is isomorphic to a sublattice characterizedby Hasse diagram in Fig. 1, then there is no idempotent nullnorm on L with the zero element a . (cid:28595) a q p a p q a p q a q p a q p a q p Figure 1: The lattices in Corollary 1
Proof.
It follows directly from Theorem 1.
Corollary 2.
Let ( L, ≤ , , be a bounded lattice, a ∈ L \{ , } , and I a = { p, q } such that p = q and p ∦ q . Then, we have p ∧ a = q ∧ a or p ∨ a = q ∨ a .Proof. Without loss of generality, assume that p ≤ q . This implies that ( p ∧ a ) ∨ ( q ∧ a ) = q ∧ a < a and( p ∨ a ) ∧ ( q ∨ a ) = p ∨ a > a . From [4, Theorem 3], it follows that there exists an idempotent nullnormon L with the zero element a . These, together with Theorem 1, imply that9a) p ∨ a ≤ q ∨ a and q ∧ a ≤ p ∧ a ; or(b) p ∧ a ≤ q ∧ a and q ∨ a ≤ p ∨ a .If p ∧ a = q ∧ a , the theorem is completed. Otherwise, if p ∧ a < q ∧ a , then (b) is satisfied, implying that q ∨ a ≤ p ∨ a . This, together with p ≤ q , implies that p ∨ a = q ∨ a . Corollary 3.
Let ( L, ≤ , , be a bounded lattice, a ∈ L \ { , } and I a = { p, q } such that p = q and p ∦ q . Then, the binary operation V : L × L → L defined by V ( x, y ) = x ∨ y, ( x, y ) ∈ [0 , a ] ,x ∧ y, ( x, y ) ∈ [ a, ,a, ( x, y ) ∈ ([0 , a ] × [ a, ∪ ([ a, × [0 , a ]) ,x ∨ ( y ∧ a ) , ( x, y ) ∈ [0 , a ] × I a ,y ∨ ( x ∧ a ) , ( x, y ) ∈ I a × [0 , a ] ,x ∧ ( y ∨ a ) , ( x, y ) ∈ [ a, × I a ,y ∧ ( x ∨ a ) , ( x, y ) ∈ I a × [ a, ,p ∨ q, ( x, y ) ∈ { ( p, q ) , ( q, p ) , ( p, p ) , ( q, q ) } (3.6) is an idempotent nullnorm on L with the zero element a if and only if p ∨ a = q ∨ a. Proof. Necessity.
Let the function V : L × L → L defined by the formula (3.6) be an idempotent nullnormon L with the zero element a. Without loss of generality, suppose that p ≤ q. In this case, we have V (1 , V ( p, q )) = V (1 , p ∨ q ) = V (1 , q ) = q ∨ a and V ( V (1 , p ) , q ) = V ( p ∨ a, q ) = ( p ∨ a ) ∧ ( q ∨ a ) = p ∨ a. This, together with the associativity of V , implies that p ∨ a = q ∨ a . Sufficiency.
Let p ∨ a = q ∨ a. Without loss of generality, suppose that p ≤ q . In this case, wehave ( p ∨ q ) ∧ ( p ∨ a ) ∧ ( q ∨ a ) = q ∧ ( q ∨ a ) = q. This, together with p ∨ q = q , implies that p ∨ q =( p ∨ q ) ∧ ( p ∨ a ) ∧ ( q ∨ a ). From [4, Theorem 3], it follows that the function V : L × L → L defined bythe formula (3.6) is an idempotent nullnorm on L with the zero element a. Considering a bounded lattice L , a ∈ L \ { , } and I a = { p, q } such that p = q , p ∦ q and p ∨ a = q ∨ a ,it should be pointed out that if p ≤ q (resp. q ≤ p ) , then the nullnorm V on L given by the formula(3.6) coincides with the nullnorm V (resp. V ) on L given by the formula (3.5) (resp. (3.1)).The following Corollary 4 can be proved in a manner similar to the proof of Corollary 3. Corollary 4.
Let ( L, ≤ , , be a bounded lattice, a ∈ L \ { , } and I a = { p, q } such that p = q and p ∦ q . Then, the binary operation V : L × L → L defined by V ( x, y ) = x ∨ y, ( x, y ) ∈ [0 , a ] ,x ∧ y, ( x, y ) ∈ [ a, ,a, ( x, y ) ∈ ([0 , a ] × [ a, ∪ ([ a, × [0 , a ]) ,x ∨ ( y ∧ a ) , ( x, y ) ∈ [0 , a ] × I a ,y ∨ ( x ∧ a ) , ( x, y ) ∈ I a × [0 , a ] ,x ∧ ( y ∨ a ) , ( x, y ) ∈ [ a, × I a ,y ∧ ( x ∨ a ) , ( x, y ) ∈ I a × [ a, ,p ∧ q, ( x, y ) ∈ { ( p, q ) , ( q, p ) , ( p, p ) , ( q, q ) } (3.7) is an idempotent nullnorm on L with the zero element a if and only if p ∧ a = q ∧ a . Considering a bounded lattice L , a ∈ L \ { , } , and I a = { p, q } such that p = q , p ∦ q and p ∧ a = q ∧ a ,it should be pointed out that if p ≤ q (resp., q ≤ p ), then the nullnorm V on L given by the formula(3.7) coincides with the nullnorm V (resp., V ) on L given by the formula (3.1) (resp., (3.5)). Remark 1. (1) Considering the commutativity of V in the proof of Theorem 1, the cases 2.1.1, 2.2.1,2.3.1, 3.1.1, 3.1.2, 3.2.1, 3.2.2, 3.3.1, and 3.3.2 can be verified directly. We include them here forcompleteness.(2) It should be noted that the last lattice in Fig. 1 is isomorphic to the one depicted by Hasse diagramin [7, Theorem 2]. Proposition 1.
Let ( L, ≤ , , be a bounded lattice, a ∈ L \ { , } , and I a = { p, q } with p = q . Then,( p ∨ a < q ∨ a and q ∧ a ≤ p ∧ a ) or ( p ∨ a ≤ q ∨ a and q ∧ a < p ∧ a ) if and only if there is only oneidempotent nullnorm on L with the zero element a , which is given by the formula (3.1) . roof. Necessity. Let p ∨ a < q ∨ a and q ∧ a ≤ p ∧ a . Then, for an idempotent nullnorm V on L , it followsfrom Lemma 2 that V ( p, q ) ∈ { p, q } . Suppose that V ( p, q ) = q . By the monotonicity of V , for q <
1, itholds q = V ( p, q ) ≤ V ( p,
1) = p ∨ a . This implies that q ∨ a ≤ p ∨ a , which is a contradiction, and thus V ( p, q ) = p . Therefore, by using Lemma 1, it follows that there is only one idempotent nullnorm on L with the zero element a , which is given by the formula (3.1). Otherwise, if p ∨ a ≤ q ∨ a and q ∧ a < p ∧ a ,it can be verified similarly. Sufficiency.
Let V be the only idempotent nullnorm on L given by the formula (3.1). In this case, V ( p, q ) = p . Suppose, on the contrary, that ( p ∨ a ≮ q ∨ a or q ∧ a (cid:2) p ∧ a ) and ( p ∨ a (cid:2) q ∨ a or q ∧ a ≮ p ∧ a ). From V ( p, q ) = p , it follows that V (0 , V ( p, q )) = V (0 , p ) = p ∧ a and V ( V (0 , p ) , q ) = V ( p ∧ a, q ) = ( p ∧ a ) ∨ ( q ∧ a ). This, together with the associativity of V , implies that q ∧ a ≤ p ∧ a .Furthermore, V (1 , V ( p, q )) = V (1 , p ) = p ∨ a and V ( V (1 , p ) , q ) = V ( p ∨ a, q ) = ( p ∨ a ) ∧ ( q ∨ a ). This,together with the associativity of V , implies that p ∨ a ≤ q ∨ a . From the supposition that ( p ∨ a ≮ q ∨ a or q ∧ a (cid:2) p ∧ a ) and ( p ∨ a (cid:2) q ∨ a or q ∧ a ≮ p ∧ a ), it can be verified that p ∨ a = q ∨ a and q ∧ a = p ∧ a .This, together with the proof of Theorem 1, implies that the binary operation V : L × L → L definedby the formula (3.5) is an idempotent nullnorm on L with the zero element a . This contradicts with thefact that V is the only idempotent nullnorm on L .The following Proposition 2 can be proved in a manner similar to the proof of Proposition 1. Proposition 2.
Let ( L, ≤ , , be a bounded lattice, a ∈ L \ { , } , and I a = { p, q } with p = q . Then,( p ∧ a < q ∧ a and q ∨ a ≤ p ∨ a ) or ( p ∧ a ≤ q ∧ a and q ∨ a < p ∨ a ) if and only if there is only oneidempotent nullnorm on L with the zero element a , which is given by the formula (3.5) . Proposition 3.
Let ( L, ≤ , , be a bounded lattice, a ∈ L \ { , } and I a = { p, q } such that p = q,p ∨ a ≤ q ∨ a and q ∧ a ≤ p ∧ a . In that case, the function V : L × L → L given by the formula (3.5) isan idempotent nullnorm on L with the zero element a if and only if p ∨ a = q ∨ a and q ∧ a = p ∧ a. Proof. Necessity.
Suppose that the binary operation V : L × L → L given by the formula (3.5) is anidempotent nullnorm on L with the zero element a . Then V ( p, q ) = q . This implies that V (1 , V ( p, q )) = V (1 , q ) = q ∨ a and V ( V (1 , p ) , q ) = V ( p ∨ a, q ) = ( p ∨ a ) ∧ ( q ∨ a ) = p ∨ a , and thus p ∨ a = q ∨ a bythe associativity of V . Similarly, by applying V (0 , V ( p, q )) = V ( V (0 , p ) , q ), we have q ∧ a = p ∧ a . Sufficiency.
Let p ∨ a = q ∨ a and q ∧ a = p ∧ a . According to the proof of Theorem 1, it is clear thatthe binary operation V : L × L → L given by the formula (3.5) is an idempotent nullnorm on L with thezero element a .The following Proposition 4 can be proved in a manner similar to the proof of Proposition 3. Proposition 4.
Let ( L, ≤ , , be a bounded lattice, a ∈ L \ { , } and I a = { p, q } such that p = q , p ∧ a ≤ q ∧ a and q ∨ a ≤ p ∨ a . In that case, the binary operation V : L × L → L given by the formula (3.1) is an idempotent nullnorm on L with the zero element a if and only if p ∨ a = q ∨ a and q ∧ a = p ∧ a . Corollary 5.
Let ( L, ≤ , , be a bounded lattice, a ∈ L \ { , } and I a = { p, q } with p = q . In thatcase, the binary operations both V : L × L → L and V : L × L → L given by the formulas (3.1) and (3.5) , respectively, are idempotent nullnorms on L with the zero element a if and only if p ∨ a = q ∨ a and q ∧ a = p ∧ a . From Lemma 1 and Corollary 5, we can easily observe that on a bounded lattice L with an element a ∈ L \ { , } satisfying that I a = { p, q } , p = q , p ∨ a = q ∨ a and q ∧ a = p ∧ a , there exist only twoidempotent nullnorms V : L × L → L and V : L × L → L defined by the formulas (3.1) and (3.5),respectively. Proposition 5.
Let ( L, ≤ , , be a bounded lattice, a ∈ L \ { , } , and p, q ∈ I a . If p ∦ q , p ∧ a = q ∧ a and p ∨ a = q ∨ a , then ( p ∧ a ) ∨ ( q ∧ a ) ∨ ( p ∧ q ) = p ∧ q and ( p ∨ a ) ∧ ( q ∨ a ) ∧ ( p ∨ q ) = p ∨ q .Proof. Without loss of generality, assume that p ≤ q as p ∦ q . This, together with p ∧ a = q ∧ a and p ∨ a = q ∨ a , implies that ( p ∧ a ) ∨ ( q ∧ a ) ∨ ( p ∧ q ) = ( p ∧ a ) ∨ p = p = p ∧ q, and ( p ∨ a ) ∧ ( q ∨ a ) ∧ ( p ∨ q ) = ( q ∨ a ) ∧ q = q = p ∨ q. Remark 2.
Proposition 5 shows that [20, Theorem 2] is a special case of [4, Theorems 3 and 4].11 . Conclusion
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