Constructions of Strict Left (Right)-Conjunctive Left (Right) Semi-Uninorms on a Complete Lattice

In this paper, we further investigate the constructions of fuzzy connectives on a complete lattice. We firstly illustrate the concepts of strict left (right)-conjunctive left (right) semi-uninorms by means of some examples. Then we give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation.


Introduction
Uninorms, introduced by Yager and Rybalov [1], and studied by Fodor et al. [2], are special aggregation operators that have proven useful in many fields like fuzzy logic, expert systems, neural networks, aggregation, and fuzzy system modeling (see [3][4][5][6]). This kind of operation is an important generalization of both t-norms and t-conorms and a special combination of t-norms and t-conorms (see [2]). But, there are real-life situations when truth functions cannot be associative or commutative. By throwing away the commutativity from the axioms of uninorms, Mas et al. introduced the concepts of left and right uninorms on in [7] and later in a finite chain in [8], and Wang and Fang [9][10] studied the left and right uninorms on a complete lattice. By removing the associativity and commutativity from the axioms of uninorms, Liu [11] introduced the concept of semi-uninorms, and Su et al. [12] discussed the notions of left and right semi-uninorms, on a complete lattice. On the other hand, it is well known that a uninorm (semi-uninorm) can be conjunctive or disjunctive whenever or 1, respectively. This fact allows us to use uninorms in defining fuzzy implications (see [9,11,[13][14]).
Constructing fuzzy connectives is an interesting topic. Recently, Jenei and Montagna [15] introduced several new types of constructions of left-continuous t-norms, Wang [16] laid bare the formulas for calculating the smallest pseudo-t-norm that is stronger than a binary operation and the largest implication that is weaker than a binary operation, Su et al. [12] studied the constructions of left and right semi-uninorms on a complete lattice, Su and Wang [17] investigated the constructions of implications and coimplications on a complete lattice. and Wang et al. [18][19][20] studied the relations among implications, coimplications and left (right) semi-uninorms, on a complete lattice. Moreover, Wang et al. [21][22] investigated the constructions of conjunctive left (right) semi-uninorms, disjunctive left (right) semi-uninorms, strict left (right)-disjunctive left (right) semi-uninorm, implications satisfying the neutrality principle, coimplications satisfying the neutrality principle, and coimplications satisfying the order property. This paper is a continuation of [12,[21][22]. Motivated by these works in [12,[21][22], we will further focus on this issue and investigate constructions of the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms on a complete lattice. This paper is organized as follows. In Section 2, we recall some necessary concepts about the left (right) semi-uninorms on a complete lattice and illustrate these notions by means of some examples. In Section 3, we give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary The knowledge about lattices required in this paper can be found in [23].
Throughout this paper, unless otherwise stated, always represents any given complete lattice with maximal element 1 and minimal element 0; stands for any index set.

Strict Conjunctive Left and Right Semi-Uninorms
Noting that the commutativity and associativity are not desired for aggregation operators in a number of cases, Liu [11] introduced the concept of semi-uninorms and Su et al. [12] studied the notions of left and right semi-uninorms. Here, we recall some necessary definitions and give some examples of the left and right semi-uninorms on a complete lattice.  Here,U is the semi-uninorm [11]. U is said to be strict left-conjunctive and strict rightconjunctive if U is conjunctive and for any , ( , 1) 0 0

If a binary operation U is left arbitrary
Noting that the least upper bound of the empty set is 0 and the greatest lower bound of the empty set is 1, we have for any , For the sake of convenience, we introduce the following symbols: ( ) Example 2.1 (Su et al. [12]).
where x and y are elements of L . Then , then there exists if 0

Constructing Strict Conjunctive Left and Right Semi-Uninorms
Constructing aggregation operators is an interesting work. Recently, Jenei and Montagna [15] introduced several new types of constructions of left-continuous t-norms, Su et al. [12] studied the constructions of left and right semi-uninorms on a complete lattice, and Wang et al. [21][22] investigated the constructions of conjunctive left (right) semi-uninorms and disjunctive left (right) semi-uninorms on a complete lattice. Now, we continue this work and give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation.
It is easy to verify that ( ) Proof. We only prove that statement (1) holds. If A is left arbitrary ∨ -distributive, then A is non-decreasing in its first variable, i.e., ua A is left arbitrary ∨ -distributive. Similarly, we can show that ua A is right arbitrary ∨ -distributive when A is right arbitrary ∨ -distributive.
The theorem is proved. Below, we give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation.
The theorem is proved. Similarly, for calculating the upper and lower approximation strict right-conjunctive right semi-uninorms of a binary operation, we have the following theorem.

Conclusions and Future Works
Constructing fuzzy connectives is an interesting topic. Recently, Su et al. [12] studied the constructions of left and right semi-uninorms, and Wang et al. [17-18, 20, 22] investigated the constructions of implications and coimplications on a complete lattice. In this paper, motivated by these works, we give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation.
In a forthcoming paper, we will further investigate the constructions of left (right) semi-uninorms and coimplications on a complete lattice.