The logic induced by effect algebras

Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {E}}$$\end{document}E, we investigate a natural implication and prove that the implication reduct of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {E}}$$\end{document}E is term equivalent to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {E}}$$\end{document}E. Then, we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. For effect algebras which need not be lattice-ordered, we introduce a certain kind of implication which is everywhere defined but whose result need not be a single element. Then, we study effect implication algebras and prove the correspondence between these algebras and effect algebras satisfying the ascending chain condition. We present an axiom system in Gentzen style also for not necessarily lattice-ordered effect algebras and prove that it is an algebraic semantics for the logic induced by finite effect algebras.

tization of the logic of quantum mechanics. For enabling deductions and derivations in this logic, it is necessary to introduce the connective implication. It is worth noticing that for lattice effect algebras this task was solved in a different way in Chajda et al. (2017). The problem is that though the binary operation of an effect algebra is only partial, implication should be defined everywhere. If the considered effect algebra is lattice-ordered, then implication is usually defined by x → y := x + (x ∧ y) and called Sasaki implication, see e.g., Borzooei et al. (2018), Chajda and Länger (2019), Foulis and Pulmannová (2012) and Rad et al. (2019). Properties of this implication were described in these papers, and a certain axiomatization of the corresponding logic in Gentzen style (see e.g., Blok and Pigozzi 1989) was derived in Rad et al. (2019). However, non-lattice-ordered effect algebras are more important than lattice-ordered ones. As shown in Chajda et al. (2009), any effect algebra can be completed to a basic algebra which is total. For commutative basic algebras, a Gentzen system was presented in Botur and Halaš (2009) and for basic algebras in Chajda (2015). This motivates us to find such an axiomatization also for not necessarily latticeordered effect algebras.
Another reason for introducing implication in effect algebras is to show that this connective is related to conjunction via (left) adjointness, and hence effect algebras can be con-sidered as left residuated structures; see Chajda and Länger (2019), Chajda and Länger (submitted).
We should not direct the attention of the reader to the quantum logic aspect only but also to psychology, etc. (see Aerts 2009). The reason is that the logic of finite effect algebras cannot completely cover quantum mechanics, but it can help in other areas.
The paper is organized as follows: First, we introduce the natural implication instead of the Sasaki implication. Then, we define lattice effect implication algebras having this type of implication together with one constant as fundamental operations. We prove that lattice effect algebras and lattice effect implication algebras are term equivalent and in a natural one-to-one correspondence. Then, we derive an algebraic semantics for lattice effect implication algebras. In the second part of the paper, we extend our investigations to not necessarily lattice-ordered effect algebras satisfying the ascending chain condition, in particular to finite effect algebras. Also in this case, we characterize the operation of implication in a similar way as it was done in the lattice case. Finally, we provide an algebraic semantics of effect implication algebras.
For concepts and results concerning effect algebras, the reader is referred to the monograph by Dvurečenskij and Pulmannová (2000). The following definition and lemma are taken from Dvurečenskij and Pulmannová (2000). Definition 1.1 An effect algebra is a partial algebra E = (E, +, , 0, 1) of type (2, 1, 0, 0) where (E, , 0, 1) is an algebra and + is a partial operation satisfying the following conditions for all x, y, z ∈ E: (E1) If x + y is defined, then so is y + x and x + y = y + x, (E2) (x + y) + z is defined if and only if so is x + (y + z), and in this case (x + y) On E, a binary relation ≤ can be defined by . Then, (E, ≤, 0, 1) becomes a bounded poset and ≤ is called the induced order of E. If (E, ≤) is a lattice, then E is called a lattice effect algebra. Lemma 1.2 If (E, +, , 0, 1) is an effect algebra, ≤ its induced order and a, b, c ∈ E, then iv) If a ≤ b and b + c is defined, then a + c is defined and a + c ≤ b + c, (v) If a + c and b + c are defined, then a + c ≤ b + c if and only if a ≤ b, (vi) If a ≤ b, then a+(a+b ) = b and (b +(b +a) ) = a, (vii) a + 0 = a, (viii) 0 = 1 and 1 = 0.

Lattice effect implication algebras
For lattice effect algebras (E, +, , 0, 1), the Sasaki implication (called also material implication in Blok and Pigozzi (1989)) was introduced as follows: for all x, y ∈ E. For our sake, we will consider this in the following way: for all x, y ∈ E and call this kind of implication natural implication. Note that the natural implication of x and y coincides with the Sasaki implication of y and x . The reason for this choice is that it turns out that some computations become more feasible with this natural implication than with the Sasaki one.
In the rest of our paper, the implication investigated in lattice effect algebras will be the natural one.
Theorem 2.1 Let (E, +, , 0, 1) be a lattice effect algebra and a, b, c ∈ E. Then, Proof (i) The following are equivalent: Concerning the reduct restricted to → and ∨, we can prove the following.
Summarizing the above properties, we can introduce an alter ego of a lattice effect algebra, i.e., its implication version.

Definition 2.3
A lattice effect implication algebra is an algebra (I , →, 0) of type (2, 0) satisfying the following conditions for all x, y, z ∈ I (we abbreviate x → 0 by x and 0 by 1): In order to show the correspondence between a lattice effect algebra and the implication version, we state and prove the following three theorems.
Theorem 2.4 Let E = (E, +, , 0, 1) be a lattice effect algebra with induced order ≤ and lattice operations ∨ and ∧ and put for all x, y ∈ E.
Proof The proof follows from Theorems 2.1 and 2.2. More precisely, (i)-(iii) follow from the fact that (E, ≤, 0, 1) is a bounded poset, (iv) follows from the fact that is antitone on (E, ≤), (v) from the fact that is an involution, (vi) and (vii) follow from the fact that x ∨ y is an upper bound of x and y, (viii) follows from the fact that x ∨ y is less than or equal to every upper bound of x and y, (ix) from (E1), (x) from (E2), (xi) from (viii) of Theorem 2.2 and (xii) from (iii) of Theorem 2.2.
Theorem 2.5 Let I = (I , →, 0) be a lattice effect implication algebra and put for all x, y ∈ E. Then, EL(I) := (I , +, , 0, 1) is a lattice effect algebra with induced order ≤ and lattice operations ∨ and ∧ where x ≤ y if and only if x → y = 1, for all x, y ∈ I . Then, due to (i)-(v) of Definition 2.3, (I , ≤ , , 0, 1) is a bounded poset with an antitone involution. If a + b is defined, then a ≤ b and hence b ≤ a , i.e., b + a is defined and

and in this case
holds. Now, the following are equivalent: This shows (E3). If 1 + a is defined, then 1 ≤ a whence a = 1, i.e., a = 0 showing (E4). Hence, EL(I) is an effect algebra. According to (vi)-(viii) of Definition 2.3, we have and because is an antitone involution on (I , ≤) we have according to (xi). If, conversely, a + c = b, then a → c = 1 and a → c = b and hence according to (xii) and (xi). This shows that induced order of EL(I) coincides with ≤.
Theorem 2.6 The correspondence between lattice effect algebras and lattice effect implication algebras described by Theorems 2.4 and 2.5 is one-to-one.

Axioms and rules for the logic of lattice effect algebras
By a propositional language, we understand some set L of propositional connectives. The arity of a propositional connective c ∈ L is called the rank of c. The L-formulas are built in the usual way from the propositional variables using the connectives of L. We denote the set of all L-formulas by Fm L or Fm in brief when the language L is clear from the context. By a logic in L, we mean a standard deductive system over L or, equivalently, its consequence relation L (see Blok and Pigozzi 1989 for details). We shall use the notation whenever there is no danger of confusion.
Let us mention that a system of axioms and rules for the propositional logic induced by lattice effect algebras was already presented in Rad et al. (2019). However, that system is rather complicated because it consists of five axioms and ten derivation rules. In what follows, we present another system having only three axioms and five rules. Moreover, three of these rules, namely Modus Ponens, Suffixing and Weak Prefixing, are common in many propositional calculi.
The axiom system (A1)-(A3) with the derivation rules (MP)-(R2) will be referred to as axiom system A.
As usual, for ∪ {ϕ, ψ} ⊆ Fm, the biconditional ϕ ↔ ψ is an abbreviation for ϕ → ψ and ψ → ϕ. In order to show that the system A is really an axiom system in Gentzen style for lattice effect algebras, we prove the following important properties.
Theorem 3.1 In the propositional logic L LEA , the following are provable: Thus, applying (MP) we conclude ϕ → 1.

Algebraic semantics
For a class K of L-algebras over a language L, consider the relation | K that holds between a set of identities and a single identity ϕ ≈ ψ if every interpretation of ϕ ≈ ψ in a member of K holds provided each identity in holds under the same interpretation. In this case, we say that ϕ ≈ ψ is a K-consequence of . The relation | K is called the semantic equational consequence relation determined by K. Given a deductive system (L, L ) over a language L, a class K of L-algebras is called an algebraic semantics for (L, L ) if L can be interpreted in | K in the following sense: There exists a finite system δ i ( p) ≈ ε i ( p), (δ(ϕ) ≈ ε(ϕ), in brief) of identities with a single variable p such that for all ∪ {ϕ} ⊆ Fm, Then, δ i ≈ ε i are called defining identities for (L, L ) and K.
K is said to be equivalent to (L, L ) if there exists a finite system ( p, q) of formulas with two variables p, q such that for every identity ϕ ≈ ψ, where ϕ ψ means just (ϕ, ψ) and || K is an abbreviation for the conjunction | K and | K .
According to Rasiowa (1974) and Rasiowa and Sikorski (1970), a standard system of implicative extensional propositional calculus (SIC, for short) is a deductive system (L, L ) satisfying the following conditions: • The language L contains a finite number of connectives of rank 0, 1 and 2 and none of higher rank, • L contains a binary connective → for which the following theorems and derived inference rules hold: As one can immediately see, the system (L, LEA ) fulfils all the properties of SIC. Indeed, there is no unary connective in our language, and we have the only binary connective →. Now, the first property is (c) of Theorem 3.1, the second is (MP) and the third is (a) from Theorem 3.1. To show the fifth property, we have ϕ → ψ (ψ → χ) → (φ → χ) by (Sf), and χ → λ, λ → χ (ψ → χ) → (ψ → λ) by (WPf). The rest then follows again by (a) of Theorem 3.1.
Moreover, taking ε( p) = p, δ( p) = p → p and ( p, q) = {p → q, q → p}, it is known (see Blok and Pigozzi 1989) that every SIC has an equivalent algebraic semantics with the defining identity δ ≈ ε and with the set as an equivalence system. As a consequence, we obtain Proposition 4.1 The logic (L, LEA ) is algebraizable with equivalence formulas = {p → q, q → p} and the defining identity p ≈ p → p.
In order to show that LEA is an equivalent algebraic semantics for (L, LEA ), we use the following statement (see Blok and Pigozzi 1989, Theorem 2.17).
Proposition 4.2 Let (L, L ) be a deductive system given by a set of axioms Ax and a set of inference rules Ir. Assume (L, L ) is algebraizable with equivalence formulas and defining identities δ ≈ ε. Then, the unique equivalent semantics for (L, L ) is axiomatized by the identities together with the quasiidentities Taking into account that ϕ → ϕ in (L, LEA ), we have p → p ≈ 1, i.e., ε( p) = 1. Applying the previous proposition, we are going to prove the following.

Theorem 4.3 The equivalent algebraic semantics for
(L, f LEA ) is axiomatized by the following identities and quasiidentities:

This system just corresponds to the quasivariety of lattice effect implication algebras.
Proof It can be immediately seen that any lattice effect implication algebra fulfills the above axioms. We will prove the converse, i.e., the properties listed above yield the conditions of Definition 2.3.
We conclude that, using the equivalence between lattice effect algebras and lattice effect implication algebras and Theorem 4.3, system A is an algebraic axiomatization of the logic of lattice effect algebras.

Effect implication algebras
Although lattice effect algebras are more feasible for some kinds of algebraic investigation, effect algebras which need not be lattice-ordered play a more important role in algebraic axiomatization of the logic of quantum mechanics. Hence, it is a question if also in this case we are able to derive some semantics, axioms and rules for a Gentzen-type axiomatization in a way similar to that for lattice-ordered effect algebras.
The first question is how to define the connective implication in not lattice-ordered effect algebras. In this section, we show that this can be done successfully in the case when the effect algebra (E, +, , 0) in question is finite, or, more generally, if for every x, y ∈ E there exist maximal elements in the lower cone L(x , y ).
In the following, let E = (E, +, , 0, 1) be an effect algebra with induced order ≤ such that (E, ≤) satisfies the ascending chain condition (shortly, ACC). This condition says that in (E, ≤) there do not exist infinite ascending chains. Hence, if (E, ≤) satisfies the ACC, then every nonempty subset of E has at least one maximal element. In particular, this is true in case of finite E. Now, let a, b ∈ E and A, B ⊆ E. Here and in the following, Max A denotes the set of all maximal elements of (A, ≤), we often identify {a} with a.
Moreover, we define Since (E, ≤) satisfies the ACC, we have x → y = ∅ for all x, y ∈ E. Moreover, if E is a lattice effect algebra, then the definition of → coincides with the original natural implication → since then for all x, y ∈ E.
Since every element of Max L(x , y ) is less than or equal to y , y + Max L(x , y ) is defined for each x, y ∈ E. It is worth noticing that now x → y need not be a single element of E, but a non-void subset of E. Hence, one cannot expect that such an implication will satisfy the same properties as the implication in lattice effect algebras. On the other hand, it was already successfully used by the first and third author in Chajda and Länger (submitted) in order to show that for the implication defined in this way the effect algebra in question can be organized in an operator residuated structure.
Although not all the conditions of Theorems 2.1 and 2.2 are valid for effect algebras which need not be lattice-ordered, we still can prove several characteristic properties.
Theorem 5.1 Let (E, +, , 0, 1) be an effect algebra satisfying the ACC and a, b ∈ E. Then, Proof (i) The following are equivalent: Summarizing all what was stated for implication in effect algebras, we can introduce the following concept.
Definition 5.2 An effect implication algebra is an algebra (I , →, 0) of type (2, 0) satisfying the following conditions for all x, y, z ∈ I (we abbreviate x → 0 by x and 0 by 1): and satisfying the condition that there does not exist an infinite sequence a 1 , a 2 , a 3 , . . . of pairwise distinct elements of I satisfying a n → a n+1 = 1 for all positive integers n.
Of course, every finite effect implication algebra satisfies trivially the last condition of Definition 5.2.
We are going to show that every effect algebra satisfying the ACC induces an effect implication algebra and vice versa. Proof The proof follows from Theorem 5.1. More precisely, (i)-(iii) follow from the fact that (E, ≤, 0, 1) is a bounded poset, (iv) follows from the fact that is antitone on (E, ≤), (v) follows from the fact that is an involution, (vi) follows from (E1), (vii) from (E2), and (viii) from (i) of Theorem 5.1 and from the definition of →.
Although x → y need not be a single element of the corresponding effect algebra E, we are able to prove that an effect implication algebra can be converted into an effect algebra where the partial operation + is defined in the standard way. for all x, y ∈ I . Then, due to (i)-(v) of Definition 5.2, (I , ≤ , , 0, 1) is a bounded poset with an antitone involution. If a + b is defined, then a ≤ b and hence b ≤ a , i.e., b + a is defined and a + b = a → b = b → a = b + a according to (vi) of Definition 5.2. This shows (E1). Now, a + b is defined if and only if a ≤ b , and in this case a + b = a → b. If a + b is defined, then (a + b) + c is defined if and only if a + b ≤ c , and in this case This shows (E3). If 1 + a is defined, then 1 ≤ a whence a = 1, i.e., a = 0 showing (E4). Hence, E(I) is an effect algebra satisfying the ACC whose induced order coincides with ≤.
The following theorem shows that the correspondence between an effect algebra satisfying the ACC and its corresponding effect implication algebra is almost one-to-one.