Tense like equality algebras

In this paper, first we define the notion of involutive operator on bounded involutive equality algebras and by using it, we introduce a new class of equality algebras that we called it a tense like equality algebra. Then we investigate some properties of tense like equality algebra. For two involutive bounded equality algebras and an equality homomorphism between them, we prove that the tense like equality algebra structure can be transfer by this equality homomorphism. Specially, by using a bounded involutive equality algebra and quotient structure of it, we construct a quotient tense like equality algebra. Finally, we investigate the relation between tense like equality algebras and tense MV-algebras.


Introduction and preliminaries
As a generalization of EQ-algebras defined in [16], Jenei introduced a new class of logical algebras in [13] and called it equality algebras, where the product operation in EQ-algebras is replaced by another binary operation smaller or equal to the original product. An equality algebra consists of two binary operations meet, equivalence and constant 1. Relations of equality algebras with the other logical algebras are studied in many works [1,2,10,11,17,18]. In [10,11,14] the authors proved that any equality algebra has a corresponding BCK-meet-semilattice and any BCK-meet-semilattice with distributivity property has a corresponding equality algebra. In [17] the authors have proved that there are relations between the equality algebras and some of other logical algebras, such as residuated lattice, MTL-algebra, BL-algebra, MV-algebra, Hertz-algebra, Heyting-algebra, Boolean-algebra, EQ-algebra, and Hoop. Moisil introduced the n-valued Lukasiewicz-Moisil algebras in [15]. After him, Chang defined the MV-algebras as algebraic structures for the infinite valued Lukasiewicz propositional calculus in [7]. The propositional calculus by adding two unary operators G, H which are called tense operators and the derived operators F := − G − and P := − H − , where " − " denotes the classical negation connective is called tense logic, for more details we refer to [5]. Tense operators for MV-algebras and Lukasiewicz-Moisil algebras were introduced in [12] and studied in [4,8]. Tense operators in lattice effect algebras were introduced in [6]. Tense operators are in certain sense quantifiers which quantify the time dimension of the logic under consideration. The semantical interpretation of these tense operators G and H is as follows. Consider a pair (T, ≤), where T is a non-void set and ≤ is a partial order on T . Let x ∈ T and f (x) be a formula of a given logical calculus. We say that G(f (t)) is valid if for any s ≥ t the formula f (s) is valid. Analogously, H(f (t)) is valid if f (s) is valid for each s ≤ t. Thus the unary operators G and H constitute an algebraic counterpart of the tense operations "it is always going to be the case that" and "it has always been the case that", respectively." This concept was generalized for the so-called basic algebras in [3], where the authors have related the basic algebras with tense operators to the quantum structures which are the so-called dynamic effect algebras. Operators such as state and equality homomorphism on equality algebras are studied in [11,18]. In Section 3, we define two unary operators that we call them tense like operators and tense like equality algebras. We study on these new algebras and their properties. The last section deals with relations of the tense like equality algebras with tense MV-algebras and vice versa.

Preliminaries
In this section, we give the basic definitions and results related to equality algebras and MV-algebras which we will use in the next sections. Also, we prove some properties of equality algebras.
We write x ≤ y if and only if x ∧ y = x, as usual. Define the following two derived operations, the implication and the equivalence operation of the equality algebra X, ∼, ∧, 1 as follows, Proposition 2.2. [13]Let E = X, ∼, ∧, 1 be an equality algebra. Then the following statements hold, for all x, y, z ∈ X: Proposition 2.3. [11]In any equality algebra E = X, ∼, ∧, 1 the following statements hold, for all x, y ∈ X: Definition 2.4. [17] Let E = X, ∼, ∧, 1 be an equality algebra. Then it is called commutative if, for any x, y ∈ X, At last we bring the definition and some useful properties of MV-algebras from [12].
Definition 2.5. [12] An MV-algebra X, ⊕, −, 0 is a set X equipped with a binary operation ⊕, a unary operation −, and a distinguished constant 0 that, for any x, y, z ∈ X, satisfying Let X be an MV-algebra such that 1 = 0 − . For any x, y ∈ X, define the operations and on X by Now, we define a partial order ≤ on MV-algebra X, ⊕, −, 0 by x ≤ y if and only if x − ⊕ y = 1, for any x, y ∈ X.
Lemma 2.6. [9]Let X, ⊕, −, 0 be an MV-algebra and ≤ be the natural order. Then, for any x, y, z ∈ X, the following statements hold:

Tense like equality algebras
In this section, we introduce an operator on involutive equality algebras which we call tense like operator. Some essential properties of tense like operators and some examples are given. Then, using it we introduce a new class of equality algebras which we called it a tense like equality algebra. An equality algebra E = X, ∼, ∧, 1 is called bounded if there exists an element 0 ∈ X such that x ≥ 0, for all x ∈ X. In a bounded equality algebra E, the negation on E is a map : X → X defined by x = x → 0 = x ∼ 0, for all x ∈ X. If (x ) = x, for all x ∈ X, then E = X, ∼, ∧, 1 is called involutive. Some results related to involutive equality algebras are obtained in [17]. We say that, for any x, y ∈ X, the negation " " on E satisfies if x ≤ y, then y ≤ x (N).
Definition 3.1. Let E = X, ∼, ∧, 0, 1 be a bounded involutive equality algebra and G : X → X be a map. We call G an involutive operator if Example 3.2. (i) Let E = X, ∼, ∧, 1 be a linearly ordered equality algebra and G be an identity map on X. Then G is an involutive and increasing operator.
(ii) Let X = {0, a, b, 1} be a set such that 0 < a < b < 1. Then, for any x, y ∈ X, define the operations ∧ and ∼ on X by x ∧ y = min{x, y} and Then, by routine calculations, we can see that E = X, ∼, ∧, 1 is an equality algebra. Now, define G 1 , G 2 : X → X by Thus, it is easy to see that G i (1 ≤ i ≤ 2) is an involutive and increasing operator on X.
(iii) Let X = {0, a, b, c, d, 1} be a set such that 0 < a < b < c < d < 1. Then, for any x, y ∈ X, define the operations ∧ and ∼ on X by x ∧ y = min{x, y} and By routine calculations, we can see that E = X, ∼, ∧, 1 is an equality algebra. Now, (iv) Let X = [0, 1] and E = X, ∼, ∧, 1 be an equality algebra, where x ∼ y = 1 − |x − y| and x ∧ y = min{x, y}, for any x, y ∈ X. It is easy to see that E is an ivolutive equality algebra. Define G : X → X by G(x) = |1 − |1 − 3x||, for any x ∈ X. Then G is an involutive and decreasing operator.
Let E = X, ∼, ∧, 0, 1 be a bounded involutive equality algebra and G : X → X be an involutive operator on X, we define the kernel of G as 1} be a lattice with the diagram as given below. Define the operations ∼ and → on X by Remark 1. Let E = X, ∼, ∧, 0, 1 be a bounded involutive equality algebra and G : X → X be an involutive operator. Then ker G is not equality subalgebra of X, in general. For example in the above example, c, b ∈ ker for all x, y ∈ X, then x ∼ y ∈ ker G, whenever x, y ∈ ker G. Since 1 ∈ ker G, ker G is an equality subalgebra of E = X, ∼, ∧, 1 .
Definition 3.4. Let E = X, ∼, ∧, 0, 1 be a bounded involutive equality algebra. A tense like equality algebra is a tripled (E, G, H), where G, H : X → X are unary operators on X and G is an involutive operator on X and for any x, y ∈ X, the following hold: Example 3.5. (i) Let E = X, ∼, ∧, 1 be a linearly ordered equality algebra and G = H be identity maps on X. Then (E, G, H) is a tense like equality algebra. (ii) Let E = X, ∼, ∧, 1 be an equality algebra and G 1 : X → X be an involutive and increasing operator on X as Example 3.2(ii). If we define the map H : X → X by H) is a tense like equality algebra. Also, ker G 1 and ker H are subalgebras of E = X, ∼, ∧, 1 , where Then by routine calculations, we can see that G is an involutive operator and satisfies in the axioms (T0)-(T3). If we take H = G, then, clearly the axiom (T4) holds. Hence, (E, G, H) is a tense like equality algebra. Now, we give some properties of tense like equality algebras. Proposition 3.6. Let E = X, ∼, ∧, 1 be an involutive equality algebra. Then, for any x, y ∈ X, the following statements hold: and so x ≤ y. The proof of the converse is clear. Now, assume that x ≤ y. Then, by Proposition 2.2(viii), we have x ∼ y = x → y. Hence, by (T2) and Proposition 2.2(iii), the proof is clear.
for every x ∈ X. Since condition (N) holds, the proof is clear.
(iv) By Proposition 2.2(i), x ∼ y ≤ x → y and, by Proposition 2.3(v) and (vi), for any x, y, z ∈ X, By these facts and (T2), we have (v) By (T1) and Proposition 2.2(i), for any x, y ∈ X we have, Again, by the axiom (T1) and Proposition 2.2(vii), for any x, y ∈ X we have H(G(x) → G(y)) ≥ HG(y).
Remark 2. The obtained results in the implications (i) and (vi) of Proposition 3.6, hold for H, too.
[17]Let E = X, ∼, ∧, 0, 1 be a bounded lattice equality algebra with the negation " ". Then the following properties hold, for all x, y ∈ X, Proposition 3.8. Let (X, ∼, ∧, 0, 1) be a tense like equality algebra, where X is an involutive lattice. Then, for any x, y ∈ X, Proof. For any x, y ∈ X, by Proposition 3.7(iii) and (T1), we have Definition 3.9. [11] Let E = X, ∼, ∧, 1 be an equality algebra. A subset D ⊆ X is called a deductive system of X if, for any x, y ∈ X: The set of all deductive systems of X is denoted by DS(X).
[11]Every deductive system of an equality algebra E is a subalgebra of E.
A homomorphism between two equality algebras is defined in [18], where the authors are investigated its relation with state morphisms. If the above equality algebras are bounded, then f (0 A ) = 0 B .
We give the basic properties of bounded equality homomorphisms as follows.
be two bounded equality algebras and f : X A → X B be a bounded equality homomorphism. Then the following statements hold: ( (iv) ker f is a proper deductive system of X A . ( By the following theorem, we show that if there exists an equality homomorphism between two equality algebras, it can transfer tense like equality algebras. be two involutive bounded equality algebras and f : X A → X B be an equality homomorphism. If (E A , G A , H A ) is a tense like equality algebra, then there are mappings Thus, the axiom (T0) holds. Moreover, for any a, b ∈ X A , we have Thus, for any x, y ∈ Y , there are a, b ∈ X A such that f (a) = x, f (b) = y and we have Now, we show that G is an involutive operator. Let " " be a negation on X A and " − " be a negation on X B . Suppose x ∈ Y such that f (a) = x, for some a ∈ X A . Then Therefore, (f (X A ), G, H) is a tense like equality algebra.
be two involutive bounded equality algebras and f : X A → X B be a surjective equality homomorphism. If (E A , G A , H A ) is a tense like equality algebra, then there are mappings G, H : X B → X B such that (X B , G, H) is a tense like equality algebra.
be two involutive bounded equality algebras and f : X A → X B be an injective equality homomorphism. Then, for any x, y ∈ f (X A ), the left inverse of f satisfies Proof. The proof is a straightforward.
be two involutive bounded equality algebras and f : X A → X B be an injective equality homomorphism. If (E B , G B , H B ) is a tense like equality algebra, then there are mappings G, H : X A → X A such that (E A , G, H) is a tense like equality algebra.
Proof. It is clear that f (X A ) is a subalgebra of X B . If (E B , G B , H B ) is a tense like equality algebra, then (f (X A ), G B , H B ) becomes a tense like equality algebra. Now, for any x ∈ X A , define G, H : f (a))).
Now, we show that G is an involutive operator. Let " " be negation on X A and " − " be negation on X B . Then by Lemma 3.14(ii), we have Therefore, (X A , G, H) is a tense like equality algebra.
If an equality homomorphism between two equality algebras is injective and surjective, then it is called an equality isomorphism.
Definition 3.19. [11] Let E = X, ∼, ∧, 1 be an equality algebra. A subset Θ ⊆ X × X is called a congruence of X if it is an equivalence relation on X and, for any x 1 , y 1 , x 2 , y 2 ∈ X such that (x 1 , y 1 ), (x 2 , y 2 ) ∈ X × X, the following statements hold: The set of all congruences of X is denoted by Con(X). Let E = X, ∼, ∧, 1 be an equality algebra and D ∈ DS(X). Denote where x/Θ D = {y ∈ X : (x, y) ∈ Θ D }. We define the operations ∼, → and ∧ on X/Θ D as Hence, E/Θ D G,H = X/Θ D G,H , ∼, ∧, 1/ D G,H is involutive. Also, for all x, y ∈ X,  Finally, we show that G is an involutive operator. For all x/Θ D G,H ∈ X/Θ D G,H , Hence, (E/Θ D G,H , G, H) is a tense like equality algebra.

Relation between tense like equality algebras and tense MV-algebras
In this section, we show that there is a connection between tense MValgebras and tense like equality algebras.
In [17], the authors showed that there is a connection between equality algebras and MV-algebras as follows. is a bounded commutative equality algebra, where → and the top element 1 are defined by, x → y = x − ⊕ y and 1 = 0 − , for all x, y ∈ X. Moreover, the equivalence operation ↔ is defined by x ↔ y = (x → y) ∧ (y → x) and x → y = x ↔ (x ∧ y).

Conclusion
In this paper, we have considered equality algebras, introduced involutive operators and tense like equality algebras. We investigated relations between these new notions with the notion of tense MV-algebras. Defining involutive operators on other logical algebras such as hoop algebras and considering these algebras as tense like algebras could be topics for our next task.