The Hartman--Mycielski construction in topological MV-algebras

Abstract: Recently, topological MV-algebras have been investigated by several mathematicians. In this paper, we mainly show that for every Hausdorff topological MV-algebra $A$, there exists a natural topological isomorphism $i_A:A\rightarrow A^\bullet$ of $A$ onto a closed subalgebra of the pathwise connected, locally pathwise connected topological MV-algebra $A^\bullet$. Furthermore, we show that there is an extension to a bounded continuous function on the MV-algebra $A^\bullet$ for each continuous real-valued bounded function on a topological MV-algebra $A$. Finally, we prove that if $\varphi:A_1\rightarrow A_2$ is a continuous homomorphism of topological MV-algebras, then $\varphi$ admits a natural extension to a continuous homomorphism $\varphi^\bullet:A_1^\bullet\rightarrow A_2^\bullet$; in addition, if $\varphi$ is open and onto, then so is $\varphi^\bullet$.

• The paper establishes that every Hausdorff topological MV-algebra embeds as a closed subalgebra of a path-connected MV-algebra.
• It adapts the Hartman–Mycielski construction to ensure continuity in operations, metrics, and function extensions.
• The work preserves categorical properties, commutes with quotients, and generalizes classical results from topological group theory.

The Hartman–Mycielski Construction in Topological MV-Algebras

Introduction and Motivation

The paper investigates the interplay between algebraic and topological structures within MV-algebras by examining the Hartman–Mycielski construction. This construction has deep roots in the theory of topological groups and is leveraged here to extend MV-algebras, which serve as algebraic models for Łukasiewicz's many-valued logic, into a richer topological context. The central question addressed is whether every Hausdorff topological MV-algebra can be realized as a closed subalgebra of a pathwise connected, locally pathwise connected topological MV-algebra, analogous to the classical embedding result for topological groups.

Preliminaries: MV-Algebras and Topology

MV-algebras $(A, \oplus, *, 0)$ are commutative monoids with additional operations designed to capture logical connectives. The paper recapitulates standard facts about MV-algebras, including the lattice structure induced by the algebra, the definition of ideals, quotient MV-algebras, and the properties of homomorphisms. Important is the notion of a topological MV-algebra, where $\oplus$ and $*$ are required to be continuous with respect to the topology, and the topology is fully characterized by $0$-neighborhoods.

Hartman–Mycielski Construction for MV-Algebras

The Hartman–Mycielski construction is adapted for MV-algebras by forming $A^\bullet$, the algebra of step functions from $J = [0, 1)$ into $A$, with pointwise operations:

• $(f \oplus g)(r) = f(r) \oplus g(r)$,
• $f^*(r) = (f(r))^*$.

The construction is shown formally to yield an MV-algebra (Proposition 3.2), retaining all defining axioms. The topology on $A^\bullet$ is generated by sets $\oplus$0, where $\oplus$1 is Lebesgue measure. This topology ensures $\oplus$2 is a topological MV-algebra (Proposition 3.3), with continuity of operations preserved.

Embedding and Connectedness Properties

A key result (Theorem 3.1) establishes that every Hausdorff topological MV-algebra $\oplus$3 admits a natural topological embedding $\oplus$4, where $\oplus$5 is the constant function $\oplus$6, and the image is a closed subalgebra. The construction ensures $\oplus$7 is both pathwise connected and locally pathwise connected (Proposition 3.4), substantially generalizing the connectivity property from groups to MV-algebras.

Extensions of Metrics and Functions

The paper proves (Theorem 3.2) that every continuous bounded pseudometric $\oplus$8 on $\oplus$9 extends canonically to $*$0 via integration over step functions. If $*$1 is a metric generating the topology, the extension $*$2 likewise generates the topology of $*$3. Moreover, every continuous real-valued bounded function $*$4 on $*$5 extends to $*$6 by averaging over step functions (Corollary 3.1). This validates the functorial nature of the construction for metrics and function spaces.

Functorial Extension of Homomorphisms

A continuous homomorphism $*$7 of topological MV-algebras naturally induces $*$8 by applying $*$9 pointwise (Theorem 3.3). The extension respects continuity, openness, and surjectivity: if $0$0 is open and onto, so is $0$1. This is critical for categorical considerations, as the Hartman–Mycielski construction preserves open quotient maps (Lemma, Corollary).

Quotient and Commutativity Properties

The Hartman–Mycielski construction commutes with taking quotients: given an ideal $0$2, the quotient $0$3 is canonically isomorphic to $0$4 as topological MV-algebras (Corollary 3.3). The kernel of $0$5 coincides with $0$6 where $0$7. This result guarantees preservation of algebraic structure under functorial passage to $0$8 and is operational for forming quotient objects.

Strong Numerical Results and Claims

• For every Hausdorff topological MV-algebra $0$9, $A^\bullet$0 is a topological isomorphism onto a closed subalgebra of a pathwise connected, locally pathwise connected MV-algebra $A^\bullet$1.
• Every continuous bounded pseudometric and real-valued function extends canonically, with the extension preserving continuity and boundedness.
• The Hartman–Mycielski construction preserves categorical properties: openness, surjectivity, and quotients.

Implications and Future Directions

This work situates topological MV-algebras alongside classic results in topological group theory by providing an explicit embedding construction with superior pathwise connectedness properties. The categorical functoriality and commutativity with quotients open avenues for deeper analysis of homological and cohomological structures in MV-algebra theory. Practically, this construction enables robust extension of metrics and continuous functions, which is vital for analytic approaches in logical systems and fuzzy set theory.

Future developments may include:

• Investigating finer invariants of $A^\bullet$2 relative to $A^\bullet$3, including duality theory and topological entropy.
• Exploring the impact on probabilistic MV-algebras and integration theory in many-valued logic.
• Extension to broader algebraic structures such as BL-algebras or alternative logical systems.

Conclusion

The paper rigorously demonstrates that the Hartman–Mycielski construction provides a universal topological extension for MV-algebras, ensuring pathwise and local connectedness, categorical functoriality, and compatibility with algebraic quotients. These results deepen the structural understanding of MV-algebras in topology and logic, establishing a foundational tool for future theoretical development and applications in mathematical logic and fuzzy systems.

(2606.08541)