Strict Left (Right)-Conjunctive Left (Right) Semi-Uninorms and Implications Satisfying the Order Property

: We firstly give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation. Then, we lay bare the formulas for calculating the upper and lower approximation implications, which satisfy the order property, of a binary operation. Finally, we reveal the relationships between the upper approximation strict left (right)-conjunctive left (right) arbitrary ˅-distributive left (right) semi-uninorms and lower approximation right arbitrary ˄-distributive implications which satisfy the order property

Uninorms, introduced by Yager and Rybalov [11], and studied by Fodor et al. [12], are special aggregation operators that have proven useful in many fields like fuzzy logic, expert systems, neural networks, aggregation, and fuzzy system modeling. 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. 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 [0, 1] in [13] and later in a finite chain in [14], and Wang and Fang [8,15] studied the left and right uninorms on a complete lattice. By removing the associativity and commutativity from the axioms of uninorms, Liu [9] introduced the concept of semi-uninorms, and Su et al. [16] 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) U can be conjunctive or disjunctive whenever (0, 1) 0 U = or 1, respectively. This fact allows us to use uninorms in defining fuzzy implications. Constructing fuzzy connectives is an interesting topic. Recently, Jenei and Montagna [17] introduced several new types of constructions of left-continuous t-norms, Wang [18] 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. [16] studied the constructions of left and right semi-uninorms on a complete lattice, Su and Wang [19] investigated the constructions of implications and coimplications on a complete lattice. and Wang et al. [20][21][22] studied the relations among implications, coimplications and left (right) semi-uninorms, on a complete lattice. Moreover, Wang et al. [23][24] investigated the constructions of conjunctive left (right) semi-uninorms, disjunctive left (right) semi-uninorms, strict left (right)-disjunctive left (right) semi-uninorm, implications and coimplications satisfying the neutrality principle.
This paper is a continuation of [16,19,[23][24]. Motivated by these works in [16,19,[23][24], we will further focus on this issue and investigate constructions of the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms and the upper and lower approximation implications which satisfy the order property.
This paper is organized as follows. In Section 2, 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 Section 3, we lay bare the formulas for calculating the upper and lower approximation implications, which satisfy the order property, of a binary operation. In Section 4, we reveal the relationships between the upper approximation strict left (right)-conjunctive left (right) arbitrary ∨ -distributive left (right) semi-uninorms and lower approximation right arbitrary ∧ -distributive implications which satisfy the order property, and find out some conditions such that the lower approximation strict left (right)-conjunctive left (right) semi-uninorm of a binary operation and upper approximation implication, which satisfies the order property, of left (right) residuum of the binary operation satisfy the generalized modus ponens rule.
The knowledge about lattices required in this paper can be found in [25].
Throughout this paper, unless otherwise stated, L always represents any given complete lattice with maximal element 1 and minimal element 0; J stands for any index set.

Strict Conjunctive Left and Right Semi-Uninorms
In this section, we firstly recall some necessary concepts about the strict conjunctive left (right) semi-uninorms on a complete lattice.
If a binary operation U is left arbitrary ∨ -distributive ( ∧distributive) and also right arbitrary For the sake of convenience, we introduce the following symbols: ( )    [17] introduced several new types of constructions of left-continuous t-norms, Su et al. [16] studied the constructions of left and right semi-uninorms on a complete lattice, and Wang et al. [23][24] 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 ( ) is the smallest strict left-conjunctive left semi-uninorm that is stronger than A on L , we call it the upper approximation strict left-conjunctive left semi-uninorm of A and write as [ ) L se cs Proof. We only prove that statement (1) holds. If A is left arbitrary ∨ -distributive, then A is non-decreasing in its first variable, 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.

Implications Satisfying the Order Property
Recently, Su and Wang [19] have studied the constructions of implications and coimplications and Wang et al. [23][24] further investigated the constructions of implications and coimplications satisfying the neutrality principle on a complete lattice. This section is a continuation of [19,[23][24]. We will study the constructions of the upper and lower approximation implications which satisfy the order property.  Note that for any implication I on L , due to the monotonicity, the absorption principle holds, i.e., (0, ) ( , 1) 1 I x I x = = for any x L ∈ .
For the sake of convenience, we introduce the following symbols: ( ) Here  Proof. We only prove that statement (1) holds. Assume that A is a right arbitrary ∨ -distributive binary operation on L . Clearly, ua A is also right arbitrary ∨ -distributive. By Definition 3.3, the monotonicity of A and .
Similarly, we have that If u x ≥ , let The theorem is proved. Below, we give out the formulas for calculating the upper and lower approximation implications which satisfy the order property.

The Relations Between Strict (Right)-Conjunctive Left (Right) Semi-Uninorms and Implications
In this section, we reveal the relationships between the upper approximation strict left (right)-conjunctive left (right) arbitrary ∨ -distributive left (right) semi-uninorms and lower approximation right arbitrary ∧ -distributive implications which satisfy the order property.
The theorem is proved. Finally, we give out some conditions such that the lower approximation strict left (right)-conjunctive left (right) semi-uninorm of a binary operation and upper approximation implication, which satisfies the order property, of left (right) residuum of the binary operation satisfy the GMP rule.

Conclusions and Future Works
Constructing fuzzy connectives is an interesting topic. Recently, Su et al. [16] studied the constructions of left and right semi-uninorms, and Wang et al. [19-20, 22, 24] 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; lay bare the formulas for calculating the upper and lower approximation implications, which satisfy the order property, of a binary operation; reveal the relationships between the upper approximation strict left (right)-conjunctive left (right) arbitrary ∨ -distributive left (right) semi-uninorms and lower approximation right arbitrary ∧ -distributive implications which satisfy the order property.
In a forthcoming paper, we will further investigate the constructions of left (right) semi-uninorms and coimplications on a complete lattice.