Riesz–McNaughton functions and Riesz MV-algebras of nonlinear functions
Introduction
In this paper we focus on Riesz MV-algebras. Recall that MV-algebras are the structures corresponding to Łukasiewicz many valued logic, in the same sense in which Boolean algebras correspond to classical logic (see [3]). Riesz MV-algebras are MV-algebras enriched with an action of the interval , which makes them appealing for applications in real analysis.
The importance of Riesz MV-algebras is witnessed, for instance, by the fact that they are related to lattice ordered real vector spaces (Riesz spaces), which are object of a vast literature in functional analysis, see e.g. [1], [5], [15], [17], [22], [25], [29]. The relation is established as follows. Recall that Mundici (see e.g. [4]) established a categorial equivalence between MV-algebras and lattice ordered abelian groups with strong unity. One has also the Riesz MV-algebra version of this equivalence, to the effect that there is a categorial equivalence between Riesz MV-algebras and Riesz spaces with a strong unity. These equivalences allow one to transfer many properties from a side of the equivalence to the other. Either side has its advantages: on Riesz MV-algebras, which form a variety, we can apply universal algebra, whereas on Riesz spaces with strong unit (which do not form a variety) we can apply group theory and we are free from the “truncation” problems which occur in Riesz MV-algebras.
Usually free MV-algebras and Riesz MV-algebras (in particular the finitely generated ones) are represented by piecewise linear functions. But for applications it could be interesting to represent (Riesz) MV-algebras with nonlinear functions. One could relax the linearity requirement and consider piecewise polynomial functions, which are important for several reasons, for instance they are the subject of the celebrated Pierce–Birkhoff conjecture, see [2], and include, in one variable, the spline functions, a kind of functions which has been deeply studied, see [26] and [27]. Other examples are Lyapunov functions used in the study of dynamical systems, see [19], and logistic functions, see [28].
In this paper we push forward the idea of Riesz MV-algebras of nonlinear functions, in the same way as in a parallel paper [10] we develop the idea of MV-algebras of nonlinear functions.
In this paper we stick to continuous functions, despite that for certain applications it could be reasonable to use discontinuous functions, for instance in order to model arbitrary signals in signal processing. Continuous functions are preferable for technical reasons: for instance, they preserve compact sets, and in general, they behave well with respect to topology.
So, our Riesz MV-algebras of interest will be the Riesz MV-algebras of all continuous functions from to , which we will denote by .
An important subalgebra of is given by the Riesz MV-algebra of what we call Riesz–McNaughton functions. We call the Riesz MV-algebra of Riesz–McNaughton functions from to . That is, if it is continuous, and there are affine functions (i.e. constants or polynomials of degree one) with real coefficients, such that for every there is i with .
In other words, is the set of all piecewise affine functions with real coefficients.
As a particular case, McNaughton functions are those Riesz–McNaughton functions where coefficients are integer rather than real. We denote by the MV-algebra of McNaughton functions (it is an MV-algebra, not a Riesz MV-algebra).
The first fundamental theorem about McNaughton functions can be stated as follows:
Theorem 1.1 (See McNaughton theorem [4].) is the MV-subalgebra of freely generated by the projections , where .
The second fundamental theorem requires the notion of rational polyhedron. We call rational polyhedron a finite union of rational convex polyhedra, and we call convex rational polyhedron the convex envelope of a finite set of rational points. Now we have:
Theorem 1.2 (See [21].) The zerosets of McNaughton functions coincide with the rational polyhedra included in .
For Riesz MV-algebras we have in the literature:
Theorem 1.3 (See [11], Corollary 7.) is the Riesz MV-subalgebra of freely generated by the projections , where .
In other words, is a free Riesz MV-algebra in n generators. Then the free Riesz MV-algebras over n generators coincide with the isomorphic copies of . In this paper, we say that a structure A is a copy of a structure B when A is isomorphic to B. However, we prefer not to identify isomorphic Riesz MV-algebras of functions, because they can consist of functions with very diverse geometric properties, which may be relevant for applications.
Theorem 1.3 is a generalization of Theorem 1.1. It seems that the analogous of Theorem 1.2 for Riesz MV-algebras has never been published, so we do it in Theorem 3.3.
A possible source of applications for Riesz MV-algebras are artificial neural networks. These networks are inspired by the nervous system to process information. There exist many typologies of neural networks used in specific fields. In [9] we have a connection between particular multilayer perceptrons and the truth functions of a Riesz MV-formula (the Riesz McNaughton Functions).
The main results of the paper are:
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an extension of the duality of [20] from MV-algebras to Riesz MV-algebras;
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a characterization of zerosets of Riesz–McNaughton functions by means of polyhedra (Theorem 3.3);
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a study of copies of in ;
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a duality between several interesting categories of Riesz MV-subalgebras of and closed subsets of up to R-homeomorphism (Theorem 6.4).
Section snippets
MV-algebras
We recall some notion for MV-algebras (for more information see [4]). An MV-algebra is a structure where is a monoid (necessarily commutative, see [18]) and:
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;
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;
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(Mangani's axiom).
Other useful notation is:
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;
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(we iterate the sum n times);
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;
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;
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;
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.
In every MV-algebra we have a partial order which holds if and only if there is z such that , or equivalently, . This order is always a
The duality
We wish to define a duality, analogous to [20], between the category of finitely generated, semisimple Riesz MV-algebras and the category of closed subsets of with suitable morphisms. In order to define these morphisms, we have to replace Z-maps with R-maps, which are tuples of Riesz–McNaughton functions, rather than tuples of McNaughton functions. Likewise, Z-homeomorphisms must be replaced by R-homeomorphisms, which are invertible R-maps.
In analogy with Theorem 2.2 we have:
Theorem 3.1 There is a
Examples of Riesz MV-algebras
Before going into further technicalities, let us consider some examples.
Consider the function seen as a function from to . Clearly, is not an element of , because, for instance, its second derivative is nonzero everywhere. So, does not generate as a Riesz MV-subalgebra of . However, since is a homeomorphism of , generates a copy of in . By Theorem 1.3, this copy consists exactly of all continuous piecewise -functions in the sense of
Riesz MV-subalgebras of
In the examples we have seen that contains continuum many copies of . More generally:
Proposition 5.1 Let be any nonconstant map. Then h generates a copy of .
Proof This follows from Lemma 3.1 by taking and since is a segment of which is R-homeomorphic to . □
Of course, every constant function generates a Riesz MV-algebra isomorphic to which cannot contain any copy of (e.g. because is totally ordered, whereas is not totally ordered).
Riesz MV-subalgebras of for
We have seen that contains continuously many copies of . In fact it is enough to consider the Riesz MV-algebras generated by with . Likewise in n dimensions we can consider the Riesz MV-algebras generated by the n-tuples and we obtain:
Corollary 6.1 contains continuously many copies of .
Conclusion
As expected, the duality of [20] extends quite smoothly from MV-algebras to Riesz MV-algebras. In fact we can say that the theory of Riesz–McNaughton functions is somewhat “easier” than the theory of McNaughton functions, e.g. the proof of Theorem 1.1 in [21] requires a sophisticated analysis of rational triangulations of the cube, whereas the proof of the corresponding theorem for Riesz MV-algebras (Theorem 3.3) relies on elementary linear algebra considerations.
On the other hand, Hopfianity
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2018, Annals of Pure and Applied LogicCitation Excerpt :An MV-algebra (a Riesz MV-algebra, a DMV-algebra) is finitely generated if it is generated by a finite set of elements, and it is finitely presented if it is the quotient of a free finitely generated MV-algebra (Riesz MV-algebra, DMV-algebra) by a principal ideal. Note that free algebras and finitely presented algebras are semisimple, see [9] for the case of MV-algebras, [21] for the case of DMV-algebras and [13] for the case of Riesz MV-algebras. Finitely presented structures are intimately connected with logic since, up to isomorphism, they are Lindenbaum–Tarski algebras corresponding to finitely axiomatizable theories with finitely many variables [31, Lemma 3.19 and Theorem 6.3].
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