Compatible operations on commutative weak residuated lattices

Compatibility of functions is a classical topic in Universal Algebra related to the notion of affine completeness. In algebraic logic, it is concerned with the possibility of implicitly defining new connectives. In this paper, we give characterizations of compatible operations in a variety of algebras that properly includes commutative residuated lattices and some generalizations of Heyting algebras. The wider variety considered is obtained by weakening the main characters of residuated lattices (A, ∧, ∨, ·, →, e) but retaining most of their algebraic consequences, and their algebras have a commutative monoidal structure. The order-extension principle a ≤ b if and only if a → b ≥ e is replaced by the condition: if a ≤ b, then a → b ≥ e. The residuation property c ≤ a → b if and only if a · c ≤ b is replaced by the conditions: if c ≤ a → b , then a · c ≤ b, and if a · c ≤ b, then e → c ≤ a → b. Some further algebraic conditions of commutative residuated lattices are required.


Introduction
The problem of adding connectives to extend a logic in a "natural" way has been broadly studied. For intuitionistic calculus, the paper [4] of X. Caicedo and R. Cignoli emphasizes the algebraic aspect of the problem through the notion of compatible function, which translates to the notion of compatible connective in intuitionistic logic. Results in [4] are extended to algebraizable logics by X. Caicedo in [2] (see also [3]).
In [5], compatible functions were studied in commutative residuated lattices, following basically the characterization of compatible functions by means of the relationship between congruences and convex subalgebras ( [12]). In [16], compatible functions were studied in the weak Heyting algebra (A, ∧, ∨, →, 0, 1), which satisfy the inequality a ∧ (a → b) ≤ b, using essentially the description of compatible functions by means of the relationship between congruences and open filters [7]. In the present work, we study compatible functions in a new variety that includes the previous ones, providing a common framework to the results given in [5,16].
For a ∈ A and n ≥ 1, we define inductively a 0 = e and a n = a · a n−1 . We also define 0 (a) = a, (a) = e → a and the iterated operator n in the usual way. The map preserve finite meets, so in particular is monotonic.

Definition 1.2.
A commutative weak residuated lattice, CWRL for short, is a GCRL that satisfies the following conditions for every a, b, c ∈ A: We write CWRL for the variety of CWRLs. A commutative residuated lattice (CRL for short) is an ordered algebraic structure (A, ∧, ∨, ·, →, e), where (A, ∧, ∨) is a lattice, (A, ·, e) is a commutative monoid, and → is a binary operation such that for every a, b, c ∈ A, the condition a · b ≤ c if and only if b ≤ a → c is satisfied. The CRLs form a variety; we write CRL for this variety. It follows from properties of CRLs [12,13] that CRL is a subvariety of CWRL; in particular, the inequalities (R5), (R6) and (R7) are equalities in CRL. Moreover, if A ∈ CWRL, then A ∈ CRL if and only if (a) = a for every a ∈ A (Remark 2.5).
A subresiduated lattice is a RWH -algebra that in addition satisfies the following inequality: Subresiduated lattices were introduced by G. Epstein and A. Horn in [9], and they were also studied in [7]. The variety of subresiduated lattices will be denoted by SRL. We write H to indicate the variety of Heyting algebras.
We obtain the following diagram: The aims of this work are the following: (i) to study compatible functions in CWRL; (ii) to extend results about compatible functions in CRL [5]; (iii) to extend results about compatible functions in RWH and in SRL [16].
Note that CWRL generalizes CRL as RWH generalizes H: the conceptual step is the same. The choice of the conditions (R2)-(R7) is needed to characterize the compatible functions as a natural generalization of the case of the variety CRL. This will become more clear in the development of this work.
In Section 2, we give some basic results about the variety CWRL. In Section 3, we study the structure of the congruence lattice of any algebra of CWRL, which we shall need later. Then in Section 4, we give characterizations for compatible functions, and we prove that the variety CWRL is locally affine complete. Finally, in Section 5, we give a method to build up unary compatible functions.

Basic results
The next elemental lemmas will be important for this work. Lemma 2.1. Let A be a GCRL that satisfies (R2), and let a, b, c ∈ A.
Proof. (a): This is a direct consequence of the condition (R2).
(b): This follows from the item (a).
The following remark comes from the previous lemma.
Example 2.7. Let H be the following poset: Consider the following binary operations:

Convex subalgebras
In this section, we study the structure of the congruence lattice of any CWRL, which we shall need later in order to give characterizations for compatible functions. Since we are building on ideas of the paper [12], we recommend the reader to have that paper at hand while reading this section.
We will refer to a subset H of a commutative weak residuated lattice A as being a subalgebra of A provided H is closed with respect to the operations defined on A. Let A ∈ CWRL and let H be a convex subalgebra of A, that is, a subalgebra such that if a, b ∈ H and a ≤ c ≤ b, then c ∈ H. We write Sub C (A) for the set of convex subalgebras of A. We write Con(A) for the set of congruences of A. If θ ∈ Con(A) and a ∈ A, we write a/θ for the equivalence class of A. We will see that there is an order isomorphism between Con(A) and Sub C (A).  Lemma 3.7. Let A ∈ CWRL. Then for every n ≥ 1, the following inequalities hold: Proof. This follows from an induction based on the inequalities (R5), (R6), and (R7).
For any A ∈ CWRL, we will write A − for the negative cone of A, that is, For any S ⊆ A, we will let C[S] denote the smallest convex subalgebra containing S and will let C[a] = C [{a}]. In what follows, we will let S denote the submonoid of (A, ·, e) generated by S. We will write N for the set of natural numbers.
The following result is analogous to [12, Lemma 2.7]. It is clear that S ⊆ K ⊆ C[S]. It will suffice to show that K is a convex Thus, K is also closed under product. Finally, we show that K is closed under arrow (for it, we use Lemma 3.7 again). First, observe that a · k (h) ≤ e and k (h) ≤ b. Then we obtain a · k (h 2 ) ≤ a · k (h) · k (h) ≤ e · b = b. Thus, by Lemma 2.4, we have that Taking into account that k (h) ≤ a and b ≤ k (h) → e, we obtain that Therefore, by inequalities (3.1) and (3.2), we conclude that a → b ∈ K.
The good description of C[S] given in the previous lemma justifies the choice of the inequalities (R5), (R6), and (R7). Corollary 3.9. Let A ∈ CWRL and a ∈ A − . Then if and only if there are n, m such that n (a m ) ≤ x.

Compatible functions
In this section, we characterize the compatible functions in the variety of commutative weak residuated lattices, and we use this result to prove that the variety CWRL is locally affine complete. We start with the following.  (a 1 , . . . , a k ), f(b 1 , . . . , b k )) ∈ θ. 2. We say that f is a compatible function of A provided it is compatible with all the congruences of A.

We say that f is compatible with a congruence θ of
If A ∈ CWRL and a, b ∈ A, denote by θ(a, b) the smallest congruence that contains the element (a, b). We also define The following lemma is useful in order to give a description of compatible functions. A ∈ CWRL and a, b ∈ A. (a) If θ ∈ Con (A), then (a, b) ∈ θ if and only if d(a, b) ∈ e/θ. then (a, b) ∈ θ if and only if p(a, b) /θ(a, b) = C[d(a, b)] = C[p(a, b)].  [d(a, b)].

Lemma 4.2. Let
In a similar way, we can prove that e/θ(a, b) = C[p(a, b)]. Let H be an algebra and let f : A → A be a function. Recall the following convenient remark: f is a compatible function if and only if (f (a), f(b)) ∈ θ(a, b) for every a, b. A be a CWRL and let f : A → A be a function. The following conditions are equivalent: For every a, b ∈ A there exist n, m ∈ N with n (d(a, b) m ) ≤ d(f (a), f(b)). For every a, b ∈ A there exist n, m ∈ N with n (p(a, b) m ) ≤ p(f (a), f(b)).
The equivalence between 1 and 3 can be proved in a similar manner.
The question of whether there are compatible functions different from polynomials naturally arises. In the variety of boolean algebras, the answer is no [14], i.e., that variety is affine complete. On the other hand, the variety H of Heyting algebras is not affine complete [4]. However, H is locally affine complete in the sense that any restriction of a compatible function to a finite subset is a polynomial. Moreover, the variety CRL is locally affine complete [5,Corollary 9], and also the variety RWH [16,Corollary 7].
In the following, we prove the locally affine completeness of the variety CWRL.
where n and m are the natural numbers associated to the pair (b, x) in Remark 4.6. Then, f (x) = T x .
Proof. Let x ∈ B. For every b ∈ B, by (4.3) we have that Hence, This proves that f (x) is an upper bound of T x .
On the other hand, since n (d(x i , x i ) m ) = e for every i = 1, . . . , k, we have that n (d( It follows from the previous corollary that every finite algebra in CWRL is affine complete.

The minimum operator
In the following, we use similar ideas to those in [5,6,10] in order to study compatible functions in the variety CWRL in terms of the minimum operator.
Definition 5.1. Let A be a poset and let g : A × A → A be a function. We say that g satisfies the condition (M) if the following condition holds :   For all a, b, c ∈ A, c ≥ b implies g(a, c) ≤ g(a, b). (M) The proof of the following lemma [5, Lemma 10] follows from the fact that if A is a ∨-semilattice and g is a function that satisfies the condition (M), then g(a, g(a, b) ∨ b) ≤ g(a, b) ∨ b for every a, b ∈ A. Moreover, in this case we have that f = h.
Then we have the following characterization for unary compatible functions.