Quantum Logic Maps and Triangular Norms on D-posets

In this paper, we propose a type of generalization of triangular norms and quantum logic functions on d-poset algebra. We compare the modified constructions, and properties to the classical properties of the generalized concepts. We show several structures with proof of each type of the proposed generalization. We show several relationships that connect the triangular norms to the quantum logic maps, and also show their relationships to the classical probability space. We provide some explanatory examples that show each structure on d-poset with extra properties that depends on a definition of state.


Introduction
There are some modern trends that aim to connect various types of measurements to the algebraic concepts. One of these trends related to the concepts of triangular norms and their concepts with respect to some algebras and their properties. On the other hand, the studies of quantum logic maps show that the solutions via such maps are much effective and smoother than the classical solutions that only depend on Boolean algebra concepts. The benefit of such solutions is to find some techniques that can be used to solve the problems which are related to the quantum mechanics. One of the these solutions is related to the problem of compatibility, non-compatibility, and the properties of the correlated components under some algebraic systems. These types of studies leads to find and construct many new structures and properties, for example, constructing quantum logic maps. In this work, we propose a generalization of some quantum logic maps and triangular norms on d-poset. We also discuss some properties that associate the modified triangular norms with each other. An interpretation to the problems of compatible and non-compatible elements is presented by means of examples that show how the quantum logic maps is working. In this note we point out an explicit relationship of difference posets with structures already established in the foundation of quantum mechanics with the orthoalgebras see [8], [6], [10]. By means of state a very important notion was mention in [5] that refer to the set of automaton state. It has discussed the automaton state as portion with respect to identifiability in input/output experiments see [14,15,5]. (In 2003), Na'na'siova' has presented a notion of conditional state. Also, notions of quantum logic functions were defined and shown their structures. It has shown that probability space is not enough to describe causality, see [13]. Another essential notion that we need to mention is the notion of triangular norms. This notion was firstly occured in the fuzzy set theory [9], [2], etc. Afterwords, many other researchers presented various T-operators in fuzzy set theory, see [3,16,17]. In particular, zadeh has presented T-operator like, min and max, as specital cases of fuzzy set operators, see [18]. This work is organized by the following way: In the next section, we review the basic notions, and concepts that are related to the dposet, orthomodular lattice, othoalgebras, state quantum logic maps, and T-norms. While, the third section deals with the main ideas of this study. Finally, conclusions and remarks are presented in section four which are related to the results that we have obtained from section three.

Foundations
In this part, we review some basic concepts related to a d-poset, probability space, quantum logic functions, T-norms, and present examples that explain their characteristics and how they are implemented. Definition 2.1 [4] A D-poset, or a difference poset, is a partially ordered set L with a partial ordering ≤, greatest element 1, and partial binary operation ⊖: ‫ܮ‬ × ‫ܮ‬ → ‫,ܮ‬ called a difference, such that, for ‫,ݑ‬ ‫ݒ‬ , ‫ݓ‬ ∈ ‫,ܮ‬ ‫ݒ‬ ⊖ ‫ݑ‬ is defined if and only if ‫ݑ‬ ≤ ‫.ݒ‬ Then the following axioms hold for all ‫,ݑ‬ ‫,ݒ‬ ‫ݓ‬ ∈ ‫:ܮ‬ Another definition that need to be recalled is the definition of orthomodular lattice(OML). The importance of OML follows from the fact that all the quantum logic are illustrated on it.
Definition 2.2 [4] Let A be a lattice with the greatest element I, the smallest element O, respectively and partial ordering ≤, endowed with a unary operation ⊥: ‫ܣ‬ → ‫,ܣ‬ such that the following hold: i.
‫ݑ‬ ≤ ‫ݒ‬ implies ‫ݑ‬ ∨ ‫ݑ(‬ ୄ ∧ ‫.)ݒ‬ Then the system ℒ = ‫,ܣ(‬ ܱ, ‫,∧,∨,ܫ‬ ⊥) is said to be an orthomodular lattice (OML). One of the basic definitions that we have to review is the definition of orthoalgebra. Its properties are very necessary to illustrate our maps in an reseanable way. The necessity of existing this logic space have been proved in (d-poset and orthoalgebra are homomorphism), [4]. Definition 2.3 [4,5] An orthoalgebra is a set ‫ܮ‬ with two particular elements 0,1, and with a partial binary operation ⊕: ‫ܮ‬ × ‫ܮ‬ → ‫ܮ‬ such that for all ‫,ݑ‬ ‫,ݒ‬ ‫ݓ‬ ∈ ‫ܮ‬ we have: i. If ‫ݑ‬ ⊕ ‫ݒ‬ ∈ ‫,ܮ‬ then ‫ݒ‬ ⊕ ‫ݑ‬ ∈ ‫ܮ‬ and ‫ݑ‬ ⊕ ‫ݒ‬ = ‫ݒ‬ ⊕ ‫ݑ‬ (commutativity). ii.
If ‫ݑ‬ ⊕ ‫ݑ‬ is defined, then ‫ݑ‬ = 0 (consistency). According to the property number (iv), it shows that the following statements are true.
a. Let ‫,ݑ‬ ‫ݒ‬ ∈ ‫,ܮ‬ then ‫ݑ‬ is orthogonal to ‫,ݒ‬ written ‫ݑ‬ ⊥ ‫,ݒ‬ if and only if, ‫ݑ‬ ⊕ ‫ݒ‬ is define in ‫;ܮ‬ b. ‫ݑ‬ is less or equal to ‫ݒ‬ and written ‫ݑ‬ ≤ ‫ݒ‬ if and only if there exists an element ‫ݓ‬ ∈ ‫ܮ‬ such that ‫ݑ‬ ⊥ ‫ݓ‬ and ‫ݓ⨁ݑ‬ = ‫ݒ‬ (in this case we also write ‫ݒ‬ ≥ ‫;)ݑ‬ c. ‫ݒ‬ is the orthocomplement of ‫,ݑ‬ if and only if, ‫ݒ‬ is a (unique) element in ‫ܮ‬ such that ‫ݒ‬ ⊥ ‫ݑ‬ and ‫ݑ‬ ⊕ ‫ݒ‬ = ‫,ܫ‬ and it is written as ‫ݑ‬ ୄ .

MAICT
We notice that s-map has the following property, see [1].
Notice that ܶ * is also known as dual function to ܶ. Indeed, there are several families that hold the conditions of ܶ, and also there are others that hold ܶ * .

Generalization of T-norms and Quantum Logic Maps on D-poset
In advance, a generalization of each chosen structure of the classical t-norms, or even of the quantum logic maps needs the following necessary notations and properties that we can summarize them in the remark below. Example 3.1 Let ‫‬ be an s-map of compatible elements on the d-poset ð. Then according to [12] we have the following table of s-map.  We can, see that the values of ‫‬ ௗ in Table 1 are compatible, for example ‫‬ ௗ ‫,ݑ(‬ ‫)ݒ‬ = ‫‬ ௗ ‫,ݒ(‬ ‫,)ݑ‬ and etc.While the following example can be illustrated with an s-map that has non-compatible elements.
Example 3.2 Let ‫‬ be an s-map of Non-compatible elements on the d-poset ð . Then according to [12] we have the following table .  In particular, we can see that in Table 2, ‫‬ ௗ ‫,ݑ(‬ ‫)ݒ‬ = 0.1 , while ‫‬ ௗ ‫,ݒ(‬ ‫)ݑ‬ = 0.2. It is clear that this situation leads to non-compatible elements of s-map. Therefore, ‫‬ ௗ ‫,ݑ(‬ ‫)ݒ‬ ≠ ‫‬ ௗ ‫,ݒ(‬ ‫,)ݑ‬ and this is true for all comparable pairs in Table 2.
Next, we illustrate a dual function to s-map on d-poset ð, which its basic form was presented in the previous part.  [12], we have the following table. Table 3. Compatible elements of j-map. While a non-compatible case can be shown in the following example. Example 3.4 Let ‫ݍ‬ be an j-map of non-compatible elements on the d-poset ð . Then according to [12], we have the following table. Table 4. Non-compatible elements of j-map. In a similar way to s-map and j-map on d-poset ð, we can illustrate a definition of d-map. This map corresponds to the notion of symmetric difference. Example 3.5 Let ݀ be an d-map of compatible elements on the d-poset ð. Then according to [12], we have the following table. Table 5. Compatible elements of d-map. While the following example explains the case of non-compatible elements with dp-map. Example 3.6 Let ݀ be an d-map of Non-compatible elements on the d-poset ð. Then according to [12], we have the following table. Table 6. Non-compatible elements of d-map. In the following example, we show how the properties of Definition 3.4 are satisfied. The main property that we focus on is the property of state and how it works.
Example 3.7 Let ð be a d-poset. Let the dp-T-norm is defined by the following relation ܶ ௗ ‫,ݑ(‬ ‫)ݒ‬ = min ‫,)ݑ(ݏ(‬ ‫))ݒ(ݏ‬ (3.1) ∀ u,v ∈ ‫.ܦ‬ Then, we can see that the dp-T-norm in (3.1) fulfils the properties of Definition 3.4. In this example, it is sufficient to prove a state property because the other properties are already satisfied and trivial. So to prove ܶ ௗ (. , ‫)ܫ‬ is state, we have. In association with definitions of dp-T-norm and dp-T-conorm, we can reformulate their definitions with respect to ⊛ . This enable us to add the associative property to the definitions of dp-T-norms, and have a full properties of classical T-norms.
Definition 3.7 Let ð be a d-poset, and ܶ ௗ , ܶ * ௗ be a dp-T-norm, dp-T-conorm, respectively. Then ܶ ௗ , ܶ * ௗ are called complete dp-T-norm, and complete dp-T-conorm, respectively, if they hold all the properties of Definition 3.4, Definition 3.5, and they hold the relation, ܶ ௗ ‫ݑ(‬ ⊛ ‫,ݒ‬ ‫)ݓ‬ = ܶ ௗ ‫,ݑ(‬ ‫ݒ‬ ⊛ ‫)ݓ‬ ,where ܶ ௗ has the associative property. One of the interesting properties that we can associate it to ܶ ௗ , and ܶ * ௗ is well-known by the convex property. This can be shown by the following way