Semi-BCI Algebras

The notion of semi-BCI algebras is introduced and some of its properties are investigated. This algebra is another generalization for BCI-algebras. It arises from the "intervalization" of BCI algebras. Semi-BCI have a similar structure to Pseudo-BCI algebras however they are not the same. In this paper we also provide an investigation on the similarity between these classes of algebras by showing how they relate to the process of intervalization.

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1 Introduction

One of the most well known references on the algebraic approach to logics is the book of Rasiowa Rasiowa1974 which dates to 70s. In this book, at pages 16-17, the notion of implicative algebra, which aim at modelling a simple notion of implication is provided: An implicative algebra is an algebra of type which satisfies the following properties:

1. ,

2. If and , then ,

3. If and , then .

A direct consequence of such definition is the establishment of an order relation “” on , which is known as Order Property (OP) of implications:

 a≤b if and only if a⇒b=⊤. (1)

As a consequence, many implications are axiomatized preserving (OP). However, some interesting implications in the field of fuzzy logics do not satisfy such requirement, for example (see Bac2008Book ; PBSS18 ; Yager1980 ):

Consider the algebra , such that:

 x→\tiny YGy={1, if x=y=0yx, otherwise.

In this case , but . However, “ implies ”.

The interval counterpart of Łukasiewicz implication introduced by Bedregal and Santiago BS2013a also fails to satisfy (OP). The authors, however, revealed that the resulting implication satisfy:

1. if 111 iff ., then ;

2. if , then .

The relation “” is precisely the way-below relation Gierz2003 of the usual Kulisch-Miranker order on intervals “”. Way-below relations, “”, are auxiliary relations Gierz2003 of partial orders “”; they have the following properties:

1. if , then .

2. if , then .

3. if a smallest element exists, then .

Since (OP) connects an implication to the underlying order relation and this is connected to auxiliary relations, this paper proposes to internalize such connection through two implications; one connected to the usual partial order and the other connected to its way-below relation. The resulting algebraic structure is called semi-BCI algebra which abstracts both BCI-algebras and their intervalization.

Another generalization for BCI-algebras which contains two implications is called Pseudo-BCI algebra which was proposed by W. A. Dudek and Y. B. Jun Dudek . The connection of such algebras to Semi-BCIs is investigated here.

Since SBCIs encompass both BCIs and its interval counterpart they tend to model logics in which the notion of impreciseness is required.

The paper is organized in the following way: Section 2 provides a brief review of BCI-algebras and their properties. Section 3 provides an overview of the intervalization process. Section 4 shows the intervalization of BCI-algebras and some properties of this interval algebra. Section 5 introduces the notion of semi-BCI algebra and prove some of its properties. Section 6 discusses the relation between Semi-BCI algebras and Pseudo-BCI algebras. Finally, section 7 provides some concluding remarks.

2 BCI-Algebras

BCI-algebras are mathematical structures for modelling fuzzy logics. They were introduced by Iséki Iseki1966 in the 60’s and since then have been extensively investigated. There are several axiom systems for BCI-algebras. We will present here the axiom systems defined by Hua06 , in which he assures that the BCIs are algebras of the form which satisfy the following properties:

1. ,

2. ,

3. ,

4. and .

A BCI-algebra is called BCK-algebra if it also satisfies:

On any BCI-algebra it is possible to define a partial order “” as: “ iff ”. Therefore, a BCI-algebra is BCK if and only if is its least element.

Example 2.1

The following algebras are BCI.

1. , s.t. .

2. , where is the set difference between and .

BCI-logics interpret the Curry combinators: (B) , (C) and (I) — see hindley1986 . This set of combinators are functional counterparts for some Fuzzy Implications. They can also be interpreted by algebras: which satisfy:

1. ,

2. ,

3. ,

4. and imply .

On any such structure it is possible to define a partial order “” as:

1. iff .

Example 2.2

The following algebra satisfies properties 1-5:

1. , s.t. .

There is a way to obtain the above axioms from those of BCI-algebras and vice-versa, the correspondence can be obtained in the following way:

Proposition 2.1

Let be a BCI-algebra. The algebra , where: and satisfies the axioms 1-4.

Proof: (BCI-1). But, . The other axioms are similarly proved.

Proposition 2.2

Let be an algebra satisfying properties 1-4. The algebra , where: and is a BCI-algebra.

Proof: Analogous to Proposition 2.1.

Corollary 2.1

The relation “” is the dual partial order of “”; namely if and only if .

Terminology

Since one kind of algebra can be obtained from the other and any result obtained for one can be easily translated, by duality, to the other, both structures are called BCI-algebras. This work consider the second kind of structure for our generalization. In this context, whenever , i.e. , the BCI-algebra will be called BCK-algebra. Now, let be some properties of BCI-algebras:

Some Properties of BCI-Algebra

(for more details see Hua06 )

1. implies ,

2. implies , — (First place antitonicity)

3. implies , — (Second place isotonicity)

4. and implies ,

5. , — (Exchange)

6. implies ,

7. ,

8. , — (Left Neutrality)

9. ,

10. ,

11. .

Properties 7, 5 and 3 model the combinators B, C and I of Combinatorial Logic hindley1986 .

Proposition 2.3

Let be a BCI-algebra. is a BCK-algebra if and only if for each there exists such that and .

Proof: () Straightforward because in BCK-algebras is the greatest element, i.e. for each .

() Suppose that is not a BCK-algebra. Then, there exists such that . By hypothesis there exists such that and . So, by 8 and definition of , . Therefore, 1 fails.

As stated in the Introduction, this paper shows that the behavior of the process of intervalization does not preserve (OP). In order to precisely define what does it mean, the next section introduces the concept of intervalization over abstract partial orders.

3 Intervalization of Structures

The limited capacity of machines to store just a finite set of finitely represented objects constraints the automatic calculation (computation) of structures in which a machine representation of some objects exceeds such capacity. In the case of real numbers, although most programs provide highly accurate results, it can happen that rounding errors built up during each step in the computation produce results which are not even meaningful. For more details see the early Forsythe’s report Forsythe1970 . In 1988, Siegfried Rump Rump1988 published the result of a computed function in an IBM S/370 mainframe. The function was:

 y=333.75b6+a2(11a2b2−b6−121b4−2)+5.5b8+a2b. (2)

He calculated for and , and the result was:

1. single precision: ;

2. double precision: ;

3. extended precision: .

All results lead any user to conclude that IBM S/370 returned the correct result. However this result is WRONG and the correct result lies in the interval: . Note that even the sign is wrong!

One of the proposals to overcome this problem is due, almost simultaneously, to Ramon Moore Moore1959 ; MoorePhD and Teruo Sunaga Sunaga1958 . They developed the so-called interval arithmetic. Interval arithmetic is a set of operations on the set of all closed intervals . The operations are defined in the following way:

1. ,

2. — where ,

3. ,

4. ; provided that .

Observe that for each operation , . This reveals two important properties of this arithmetic (a) Correctness and (b) Optimality.

Correctness. The criterion for correctness of a definition of interval arithmetic is that the “Fundamental Theorem of Interval Arithmetic” holds IIIIIIMoore (Moo79, , Theorem 3.1, p. 21): If is an inclusion monotonic interval extension of , then ; where .: when an expression is evaluated using intervals, it yields an interval containing all results of pointwise evaluations based on point values that are elements of the argument intervals.

Optimality. By optimality, we mean that the computed floating-point interval is not wider than necessary.”
Hickey et.al(Hic01, , p.1040)

The philosophy behind intervals is the following: Enclosure in intervals the values which are not exact by any reason (e.g. the value comes from an imprecise measurement) and apply correct and optimal operations on such intervals in order to obtain the best interval which contains the desired output. This approach will avoid what happened with the Rump’s example. Therefore, the notion of correctness is indispensable for such philosophy.

The property of correctness was investigated in 2006 by Santiago et al BSA2013 ; SBA06 . In those papers, instead of correctness the authors used the term representation, since an interval computation could be understood not just as a machine representation of real numbers, but also as a mathematical representation of real numbers (this idea is confirmed by the Representation Theorems of Euclidean continuous functions in BSA2013 ; SBA06 ). In what follows this notion is shown for binary operations: A binary interval operation, , represents a binary real operation, , whenever:

 (x,y)∈[a,b]×[c,d]⇒x⋄y∈[a,b]⋄[c,d] (3)

This can be easily extended to -ary operations. The authors showed that this notion is more general than what is stated by the Fundamental Theorem of Interval Arithmetic; given that there are representations which are not inclusion monotonic (see (SBA06, , p. 238)).

One noteworthy point which will be taken into account in the present paper: There is a difference between the representation of a function as an interval function and an extension of a function to an interval function . For example, given intervals and , the function , presented in Markov77 , extends the subtraction on real numbers, however , , but ; in other words, this operation is not correct. So, there are interval extensions which are not correct. They are useless for the proposed philosophy.

The process of giving the correct and optimal interval version for a function is called: “intervalization”. There are many proposal of intervalization of algebraic structures further than that of real numbers proposed by Moore and Sunaga. In the literature, the reader can find proposals even for the field of Logic, since there are structures which interpret logics that are susceptible to the same situation of . For example: The Łukasiewicz implicative algebra s.t. interprets some many-valued logics and was “intervalized” by Bedregal and Santiago in BS2013a . Its MV-algebra counterpart was intervalized by Cabrer et al in Cabrer2014 , also, in order to overcome the same problems already stated for . In both cases, the interval algebras did not satisfy the same properties that are satisfied by the algebras that they came from. The same happened with !

The following section a way of “intervalizing’’ BCI-algebras is provided. Like the case of MVs and Łukasiewicz algebras the resulting structure does not belong to the same category of its starting algebra. This paper we provide an investigation of the resulting structures. In order to achieve that, some required concepts, like the abstract notion of intervals are introduced. The aim, again, is to provide the ability to use intervals to represent the elements of an algebra .

Definition 3.1 (Abstract Intervals)

Given a poset , the set is called the closed interal with endpoints and and is the set of all intervals of elements of . For any, its left and right endpoints by and , respectively, i.e. if then and . When the interval is called degenerate. The embedding , s.t. is called natural embedding. On the set it is canonical to define the partial order: if and only if and . This relation is called pointwise or Kulisch-Miranker order.

Since BCI-algebras are partially ordered systems, , it is possible to apply Definition 3.1 to obtain the partial order . The question is about the implications on : If an interval operation on satisfies the BCI axioms, is it also correct? The following section will show that the answer is negative. But what does it mean? It means it is not possible to have both: (1) correctness and (2) the known theory of BCI-algebras for . So, since correctness is indispensable, a price must be paid: A new theory for must be developed. This is the reason of this paper!

4 Intervalization of BCI-algebras

This section shows that it is not possible to have an interval BCI-algebra with a correct implication. Proposition 4.1 shows that it is possible to build an interval BCI-algebra, but with a non-correct implication, and Theorem 4.1 shows that it is an impossible task. Finally, we provide the “BCI-algebra intervalization theorem” and some properties of resulting algebra.

Lemma 4.1

Let be a BCI-algebra such that is a meet-semilattice . For each , iff and . Moreover, if and then .

Proof: Straightforward.

Lemma 4.2

Let be a BCI-algebra such that is a meet-semilattice satisfying:

 a⪯b→c iff a∧b⪯c, (4)

for every . For each , if then .

Proof: If then, by Lemma 4.1 and 5, and . By (4), and and therefore, and . So, and . Thus, applying again (4), and . Hence, . So, by (4) and 5, .

Proposition 4.1

Let be a BCI-Algebra such that is a meet-semilattice satisfying . Then , where

 X↦Y=[(X––→Y––)∧(¯¯¯¯¯X→¯¯¯¯Y),¯¯¯¯¯X→¯¯¯¯Y], (5)

is also a BCI-algebra which satisfies .

Proof: Notice that in this case, defining iff , then iff iff and .

Thus, clearly, the properties 3 to 5 are trivially satisfied. In the following, 1 is proved.

Since, is a BCI-Algebra, by 1, each , and . Then, and . So, by Lemma 4.1, . Thus, by Lemma 4.2 and Eq. (5), . Therefore by Eq. (5), (*) . On the other hand, by (C-1), and so, by Eq. (5): (**) . Thus, from (*) and (**) and Lemma 4.1, . Since, by Eq. (5), , then (***) . Thus, from (***) and (**), and . Therefore, by Eq. (5), .

2: Clearly, and therefore, by (A-2), . So, by 3 and 2, . So, by Eq. (5) and 1, . On the other hand, by Eq. (5) and 2, (#) . Therefore, (##) . Hence, from (##), (#) and Eq. (5): and . Consequently, . Therefore, by Eq. (5), .

is a meet-semilattice. In fact, let . Then, iff and iff , , and iff and .
In addition, iff and iff and iff and iff . Therefore, satisfies (4).

If has two different elements, say and , such that , i.e. , then is not an interval representation of . In particular, . Nevertheless, by 4, and so . This leads us to the following general theorem:

Theorem 4.1

Let be a BCI-Algebra. If there are such that and , then for any interval there is no interval representation for such that is a BCI-algebra.

Proof: Case . Then and therefore is not an interval representation of .

Case , then . Nevertheless, by 4, and so . Therefore, in this case also is not an interval representation of .

Case for some . Then . However, if is a BCI-algebra then, by (A-8), and therefore, which means that again is not an interval representation of .

In the following, we propose a process for intervalization of BCI-algebras.

Theorem 4.2

Let be a BCI-algebra, be a meet semilattice, such that for each , and . For , define:

1. .

Then is the best representation of and the structure satisfies:

1. ,

2. ,

3. ,

4. ,

5. ,

6. ,

7. e ,

where e . However, when has at least one element different from , then is not a BCI-algebra.

Proof: According to Proposition 4.4 at BS2013a the operation is the best representation of . Note that:

1. and and .

2. and and and and and , i.e. is the Kulisch-Miranker order.

7 is satisfied, since is a poset and “” is the Kulisch-Miranker order.

Case of 1. . According to property 5 of BCI-algebras this term is also equal to .

Case of 2. . On the other hand, . By property 5, the last term is equal to: which is equal IIIIIIIIIBy associativity and commutativity of meet. to

Case of 3. By definition, and . Since, by 2, 3 and 7, and , then . On the other hand, by 2, 3 and 7, . Therefore, .

Case of 4. By (A-8),