On the variety of equality algebras - Atlantis Press
EUSFLAT-LFA 2011
July 2011
Aix-les-Bains, France
On the variety of equality algebras∗ Sándor Jenei1 László Kóródi2 1
1
Institute of Mathematics and Informatics, University of Pécs, Hungary Department of Knowledge-Based Mathematical Systems, Johannes Kepler University, Linz, Austria 2 Institute of Mathematics and Informatics, University of Pécs, Hungary
(E5) a ≤ b ≤ c implies a ∼ c ≤ b ∼ c and a ∼ c ≤ a ∼ b,
Abstract Equality algebras has recently been introduced. A subclass of equality algebras, called equivalential equality algebras is closely related to BCK-algebras with meet. We show that the variety of equality algebras has nice properties: We shall investigate their congruences and filters and prove that the variety of equality algebras is a 1-regular, arithmetic variety.
(E6) a ∼ b ≤ (a ∧ c) ∼ (b ∧ c), (E7) a ∼ b ≤ (a ∼ c) ∼ (b ∼ c). The operation ∧ is called meet (infimum) and ∼ is an equality operation. We write a ≤ b iff a ∧ b = a, as usual and define the following two derived operations, the implication and the equivalence operation of the equality algebra E by
Keywords: Equality algebra, BCK-algebra, 1regular, congruence permutable, congruence distributive variety
a → b = a ∼ (a ∧ b) a ↔ b = (a → b) ∧ (b → a).
1. Introduction
Call an equality algebra (and as well its equality operation ∼) equivalential if ∼ coincides with ↔.
The motivation for introducing equality algebras came from EQ-algebras [6]. In EQ-algebras, compared to equality algebras, there is an additional operation ⊗, called product, which is very loosely related to the other operations. Therefore, there might not exist deep algebraic characterizations of EQ-algebras, and our intention was to define a structure similar to EQ-algebras but without the product. That has lead to the following axioms:
One can prove that equality algebras are exactly the ⊗-free subreducts of the so-called good EQ-algebras. The general motivation for equality algebras from the side of logic was to define an algebraic structure which (with appropriate extensions) is suitable to axiomatize a large class of substructural logics based on an equivalence connective rather than implication. The very first step toward this aim has been done in [4] where it has been shown that
Definition 1 An equality algebra [4] is an algebra E = hX, ∼, ∧, 1i of type (2, 2, 0) such that the following axioms are fulfilled for all a, b, c ∈ X:
1. Equality algebras form a variety. 2. The class of equivalential equality algebras and the class of BCK-algebras with meet are term equivalent.
(E1) hX, ∧, 1i is a commutative idempotent integral monoid (i.e. ∧-semilattice with top element 1),
3. A generalization of a result of Kabziński and Wroński in [5] was obtained, namely, it holds true that ∼ can be represented as the equivalence operator of a BCK-algebra with meet (on a ∧-semilattice (X, ∧) with top element 1) if and only if (E2)-(E4), (E6),
(E2) a ∼ b = b ∼ a, (E3) a ∼ a = 1, (E4) a ∼ 1 = a, ∗ This work was supported by the OTKA-76811 grant, the SROP-4.2.1.B-10/2/KONV-2010-0002 grant, and the MC ERG grant 267589.
© 2011. The authors - Published by Atlantis Press
(1) (2)
a ∼ (a ∧ b ∧ c) ≤ a ∼ (a ∧ b),
153
a ∼ (a ∧ b) ≤ (a ∧ c) ∼ (a ∧ b ∧ c),
(h) a ≤ (a → b) → b, a → ((a → b) → b) = 1,
a ∼ b = (a ∼ (a ∧ b)) ∧ (b ∼ (a ∧ b))
(i) a → b ≤ (b → c) → (a → c), (a → b) → ((b → c) → (a → c)) = 1
hold.
(j) a ≤ b → c iff b ≤ a → c,
4. All totally ordered equality algebras are equivalential.
(k) a → (b → c) = b → (a → c),
Point 3 above seems to be important since it tells us about an equational characterization of the equivalence operation of BCK semilattices, which may easily be arisen to an axiomatic description of the equivalential fragment of the related logic.
(l) a ↔ a = 1, a ↔ 1 = a, (m) b ≤ a implies a ↔ b = a → b = a ∼ b. 2. Filters and congruences in equality algebras
The term equivalence, which is mentioned above at point 2 is given by the following Theorem 1 [4] The following two statements hold true:
Next, we investigate the filter theory of equality algebras.
i. For any equality algebra E = hX, ∼, ∧, 1i, Ψ(E) = hX, →, ∧, 1i is a BCK-algebra with meet.
Definition 3 Let E = hE, ∼, ∧, 1i be an equality algebra and F ⊆ E. 1. F is called a deductive system or filter of E if for all a, b ∈ E we have
ii. For any BCK-algebra with meet B = hX, → , ∧, 1i, Φ(B) = hX, ↔, ∧, 1i is an equality algebra, where ↔ denotes the equivalence operation of B. Moreover, the implication of Φ(B) coincides with →, that is, we have a → b = a ↔ (a ∧ b).
(i) 1 ∈ F , (ii) a ∈ F, a ≤ b ⇒ b ∈ F , (iii) a, a ∼ b ∈ F ⇒ b ∈ F ,
(3)
Denote F il(E) the set of all filters of E. Clearly, F il(E) closed under arbitrary intersections and {1} ∈ F il(E), so hF il(E), ⊆i is a complete lattice. A filter F of an equality algebra E is proper if F 6= E . A proper filter F is called maximal if F ⊆ G ⊆ E implies F = G for all G proper filter of E.
Definition 2 Employing the notations of Theorem 1 call B = Ψ(E) the underlying BCK-algebra of E and call E¯ = Φ(B) the canonical equality algebra of B. To conclude this section, the basic properties of equality algebras are summarized:
2. A subset Θ of E × E is called congruence of E, if it is an equivalence relation on E and for all a, a0 , b, b0 ∈ E such that (a, b), (a0 , b0 ) ∈ Θ, it holds that
Proposition 2 [4] Let E = hX, ∼, ∧, 1i be an equality algebra. Then the followings hold for all a, b, c, d ∈ X: (a) a ∼ b ≤ a ↔ b ≤ a → b,
(i) (a ∧ a0 , b ∧ b0 ) ∈ Θ,
(b) a ≤ (a ∼ b) ∼ b,
(ii) (a ∼ a0 , b ∼ b0 ) ∈ Θ.
(c) a ∼ b = 1 iff a = b,
Denote Con(E) the set of all congruences of E.
(d) a → b = 1 iff a ≤ b,
3. For F ∈ F il(E) define the following relations on E:
(e) a → b = 1 and b → a = 1 imply a = b, (f) 1 → a = a, a → 1 = 1, a → a = 1,
− ⇐⇒ {x → y, y → x} ⊆ F, (x, y) ∈ Θ→ F
(g) a ≤ b → a, a → (b → a) = 1,
(x, y) ∈ ΘF ⇐⇒ x ∼ y ∈ F.
154
The following proposition states that the set of filters of an equality algebra coincide with the set of (BCK-algebra) filters of its underlying BCKalgebra.
[2] K. Iseki, S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon., 23 (1978), 1–26. [3] P. M. Idziak, Lattice operation in BCKalgebras, Math. Japonica (1984) vol. 29 (6) 839–846. [4] S. Jenei, Equality algebras, (submitted) [5] J. Kabziński, A. Wroński, On equivalential algebras, Proceedings of the 1975 International Symposium on Multiple-valued Logic, Indiana University, Bloomington, 1975, 231–243. [6] V. Novák, B. De Baets. EQ-algebras, Fuzzy Sets and Systems, 160 (20), (2102) 2956-2978. [7] M. Palasinski, On ideal and congruence lattices of BCK-algebras, Math., Japon., 26 (1981), 543–544.
Proposition 3 Let E = hE, ∼, ∧, 1i be an equality algebra. F ∈ F il(E) iff for all a, b ∈ E, (i’) 1 ∈ F , (ii’) a, a → b ∈ F ⇒ b ∈ F holds. Proposition 4 If E is an equality algebra and −. F ∈ F il(E) then ΘF ∈ Con(E) and ΘF = Θ→ F Lemma 5 For Θ ∈ Con(E) we have (a, b) ∈ Θ iff (a ∼ b, 1) ∈ Θ The next theorem establishes a connection between F il(E) and Con(E). Theorem 6 Let E = hE, ∼, ∧, 1i be an equality algebra, Θ, Ψ ∈ Con(E), F ∈ F il(E). Then (a) [1]Θ ∈ F il(E), where [1]Θ = {a | (a, 1) ∈ Θ}, (b) Θ[1]Θ = Θ, (c) [1]ΘF = F , (d) (1-regularity) if [1]Θ = [1]Ψ , then Θ = Ψ. Lemma 7 The variety of equality algebras is congruence permutable and congruence distributive. Remark 8 Every variety in which (E3), (E4), and (b) (or (E7) and (E2) instead of (b)) holds is congruence permutable. The term m(x, y, z) = ((x ∼ y) ∼ z) ∧ ((y ∼ z) ∼ x). testifies it. Summing up, we have obtained that Theorem 9 The variety of equality algebras is a 1-regular, arithmetical variety. References [1] N. Galatos, P. Jipsen, T, Kowalski, H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Volume 151. Studies in Logic and the Foundations of Mathematics (2007) pp. 532
155
July 2011
Aix-les-Bains, France
On the variety of equality algebras∗ Sándor Jenei1 László Kóródi2 1
1
Institute of Mathematics and Informatics, University of Pécs, Hungary Department of Knowledge-Based Mathematical Systems, Johannes Kepler University, Linz, Austria 2 Institute of Mathematics and Informatics, University of Pécs, Hungary
(E5) a ≤ b ≤ c implies a ∼ c ≤ b ∼ c and a ∼ c ≤ a ∼ b,
Abstract Equality algebras has recently been introduced. A subclass of equality algebras, called equivalential equality algebras is closely related to BCK-algebras with meet. We show that the variety of equality algebras has nice properties: We shall investigate their congruences and filters and prove that the variety of equality algebras is a 1-regular, arithmetic variety.
(E6) a ∼ b ≤ (a ∧ c) ∼ (b ∧ c), (E7) a ∼ b ≤ (a ∼ c) ∼ (b ∼ c). The operation ∧ is called meet (infimum) and ∼ is an equality operation. We write a ≤ b iff a ∧ b = a, as usual and define the following two derived operations, the implication and the equivalence operation of the equality algebra E by
Keywords: Equality algebra, BCK-algebra, 1regular, congruence permutable, congruence distributive variety
a → b = a ∼ (a ∧ b) a ↔ b = (a → b) ∧ (b → a).
1. Introduction
Call an equality algebra (and as well its equality operation ∼) equivalential if ∼ coincides with ↔.
The motivation for introducing equality algebras came from EQ-algebras [6]. In EQ-algebras, compared to equality algebras, there is an additional operation ⊗, called product, which is very loosely related to the other operations. Therefore, there might not exist deep algebraic characterizations of EQ-algebras, and our intention was to define a structure similar to EQ-algebras but without the product. That has lead to the following axioms:
One can prove that equality algebras are exactly the ⊗-free subreducts of the so-called good EQ-algebras. The general motivation for equality algebras from the side of logic was to define an algebraic structure which (with appropriate extensions) is suitable to axiomatize a large class of substructural logics based on an equivalence connective rather than implication. The very first step toward this aim has been done in [4] where it has been shown that
Definition 1 An equality algebra [4] is an algebra E = hX, ∼, ∧, 1i of type (2, 2, 0) such that the following axioms are fulfilled for all a, b, c ∈ X:
1. Equality algebras form a variety. 2. The class of equivalential equality algebras and the class of BCK-algebras with meet are term equivalent.
(E1) hX, ∧, 1i is a commutative idempotent integral monoid (i.e. ∧-semilattice with top element 1),
3. A generalization of a result of Kabziński and Wroński in [5] was obtained, namely, it holds true that ∼ can be represented as the equivalence operator of a BCK-algebra with meet (on a ∧-semilattice (X, ∧) with top element 1) if and only if (E2)-(E4), (E6),
(E2) a ∼ b = b ∼ a, (E3) a ∼ a = 1, (E4) a ∼ 1 = a, ∗ This work was supported by the OTKA-76811 grant, the SROP-4.2.1.B-10/2/KONV-2010-0002 grant, and the MC ERG grant 267589.
© 2011. The authors - Published by Atlantis Press
(1) (2)
a ∼ (a ∧ b ∧ c) ≤ a ∼ (a ∧ b),
153
a ∼ (a ∧ b) ≤ (a ∧ c) ∼ (a ∧ b ∧ c),
(h) a ≤ (a → b) → b, a → ((a → b) → b) = 1,
a ∼ b = (a ∼ (a ∧ b)) ∧ (b ∼ (a ∧ b))
(i) a → b ≤ (b → c) → (a → c), (a → b) → ((b → c) → (a → c)) = 1
hold.
(j) a ≤ b → c iff b ≤ a → c,
4. All totally ordered equality algebras are equivalential.
(k) a → (b → c) = b → (a → c),
Point 3 above seems to be important since it tells us about an equational characterization of the equivalence operation of BCK semilattices, which may easily be arisen to an axiomatic description of the equivalential fragment of the related logic.
(l) a ↔ a = 1, a ↔ 1 = a, (m) b ≤ a implies a ↔ b = a → b = a ∼ b. 2. Filters and congruences in equality algebras
The term equivalence, which is mentioned above at point 2 is given by the following Theorem 1 [4] The following two statements hold true:
Next, we investigate the filter theory of equality algebras.
i. For any equality algebra E = hX, ∼, ∧, 1i, Ψ(E) = hX, →, ∧, 1i is a BCK-algebra with meet.
Definition 3 Let E = hE, ∼, ∧, 1i be an equality algebra and F ⊆ E. 1. F is called a deductive system or filter of E if for all a, b ∈ E we have
ii. For any BCK-algebra with meet B = hX, → , ∧, 1i, Φ(B) = hX, ↔, ∧, 1i is an equality algebra, where ↔ denotes the equivalence operation of B. Moreover, the implication of Φ(B) coincides with →, that is, we have a → b = a ↔ (a ∧ b).
(i) 1 ∈ F , (ii) a ∈ F, a ≤ b ⇒ b ∈ F , (iii) a, a ∼ b ∈ F ⇒ b ∈ F ,
(3)
Denote F il(E) the set of all filters of E. Clearly, F il(E) closed under arbitrary intersections and {1} ∈ F il(E), so hF il(E), ⊆i is a complete lattice. A filter F of an equality algebra E is proper if F 6= E . A proper filter F is called maximal if F ⊆ G ⊆ E implies F = G for all G proper filter of E.
Definition 2 Employing the notations of Theorem 1 call B = Ψ(E) the underlying BCK-algebra of E and call E¯ = Φ(B) the canonical equality algebra of B. To conclude this section, the basic properties of equality algebras are summarized:
2. A subset Θ of E × E is called congruence of E, if it is an equivalence relation on E and for all a, a0 , b, b0 ∈ E such that (a, b), (a0 , b0 ) ∈ Θ, it holds that
Proposition 2 [4] Let E = hX, ∼, ∧, 1i be an equality algebra. Then the followings hold for all a, b, c, d ∈ X: (a) a ∼ b ≤ a ↔ b ≤ a → b,
(i) (a ∧ a0 , b ∧ b0 ) ∈ Θ,
(b) a ≤ (a ∼ b) ∼ b,
(ii) (a ∼ a0 , b ∼ b0 ) ∈ Θ.
(c) a ∼ b = 1 iff a = b,
Denote Con(E) the set of all congruences of E.
(d) a → b = 1 iff a ≤ b,
3. For F ∈ F il(E) define the following relations on E:
(e) a → b = 1 and b → a = 1 imply a = b, (f) 1 → a = a, a → 1 = 1, a → a = 1,
− ⇐⇒ {x → y, y → x} ⊆ F, (x, y) ∈ Θ→ F
(g) a ≤ b → a, a → (b → a) = 1,
(x, y) ∈ ΘF ⇐⇒ x ∼ y ∈ F.
154
The following proposition states that the set of filters of an equality algebra coincide with the set of (BCK-algebra) filters of its underlying BCKalgebra.
[2] K. Iseki, S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon., 23 (1978), 1–26. [3] P. M. Idziak, Lattice operation in BCKalgebras, Math. Japonica (1984) vol. 29 (6) 839–846. [4] S. Jenei, Equality algebras, (submitted) [5] J. Kabziński, A. Wroński, On equivalential algebras, Proceedings of the 1975 International Symposium on Multiple-valued Logic, Indiana University, Bloomington, 1975, 231–243. [6] V. Novák, B. De Baets. EQ-algebras, Fuzzy Sets and Systems, 160 (20), (2102) 2956-2978. [7] M. Palasinski, On ideal and congruence lattices of BCK-algebras, Math., Japon., 26 (1981), 543–544.
Proposition 3 Let E = hE, ∼, ∧, 1i be an equality algebra. F ∈ F il(E) iff for all a, b ∈ E, (i’) 1 ∈ F , (ii’) a, a → b ∈ F ⇒ b ∈ F holds. Proposition 4 If E is an equality algebra and −. F ∈ F il(E) then ΘF ∈ Con(E) and ΘF = Θ→ F Lemma 5 For Θ ∈ Con(E) we have (a, b) ∈ Θ iff (a ∼ b, 1) ∈ Θ The next theorem establishes a connection between F il(E) and Con(E). Theorem 6 Let E = hE, ∼, ∧, 1i be an equality algebra, Θ, Ψ ∈ Con(E), F ∈ F il(E). Then (a) [1]Θ ∈ F il(E), where [1]Θ = {a | (a, 1) ∈ Θ}, (b) Θ[1]Θ = Θ, (c) [1]ΘF = F , (d) (1-regularity) if [1]Θ = [1]Ψ , then Θ = Ψ. Lemma 7 The variety of equality algebras is congruence permutable and congruence distributive. Remark 8 Every variety in which (E3), (E4), and (b) (or (E7) and (E2) instead of (b)) holds is congruence permutable. The term m(x, y, z) = ((x ∼ y) ∼ z) ∧ ((y ∼ z) ∼ x). testifies it. Summing up, we have obtained that Theorem 9 The variety of equality algebras is a 1-regular, arithmetical variety. References [1] N. Galatos, P. Jipsen, T, Kowalski, H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Volume 151. Studies in Logic and the Foundations of Mathematics (2007) pp. 532
155
