Stone Duality Type Theorems for MV-Algebras with Internal State

Abstract

Recently in [10, 11], the language of MV-algebras was extended by adding a unary operation, an internal operator, called also a state-operator. In [5], a stronger version of state MV-algebras, called state-morphism MV-algebras, was given. In this article, we present Stone Duality Theorems for (i) the category of Boolean algebras with a fixed state-operator and the category of compact Hausdorff topological spaces with a fixed idempotent continuous function, and for (ii) the category of weakly divisible σ-complete state-morphism MV-algebras and the category of Bauer simplices whose set of extreme points is basically disconnected and with a fixed idempotent continuous function.

Key Words:

• Bauer simplex
• Boolean algebra
• Internal state
• MV-algebra
• State MV-algebra
• State-morphism MV-algebra
• State-morphism-operator
• Stone duality

2000 Mathematics Subject Classification:

• 06D35
• 03B50
• 03G12

ACKNOWLEDGMENTS

The article has been supported by the Center of Excellence SAS–Quantum Technologies, ERDF OP R&D Projects CE QUTE ITMS 26240120009, and meta-QUTE ITMS 26240120022, the grant VEGA No. 2/0032/09 SAV, by the Slovak Research and Development Agency under the contract APVV-0071-06, Bratislava, and by Slovak–Italian project SK-IT 0016-08.

Notes

Communicated by A. Olshanskii.