Natural duality, modality, and coalgebra.

*(English)*Zbl 1263.08002In this paper the author presents a family of topological dualities between classes of algebras with a modal operator and categories of enriched Stone spaces. The classes of algebras considered in the paper are constructed from a semi-primal algebra with a bounded lattice reduct and suitably defining modalities on its powers. Then, appropriately extending the Vietoris functor to each of the enriched Stone spaces, the author also presents a coalgebraic version of the duality.

Several authors have tackled the problem of adding modalities to some classes of algebras with lattice reduct and to obtain topological representations for these classes, e.g., R. Cignoli, S. Lafalce and A. Petrovich [Order 8, No. 3, 299–315 (1991; Zbl 0754.06006)], A. Petrovich [Order 16, No. 1, 1–17 (1999; Zbl 0952.06016)], S. Celani and R. Jansana [Log. J. IGPL 7, No. 6, 683–715 (1999; Zbl 0948.03013)], C. B. Wegener [Stud. Log. 70, No. 3, 339–352 (2002; Zbl 1007.06010)], M. Gehrke, H. Nagahashi and Y. Venema [Ann. Pure Appl. Logic 131, No. 1–3, 65–102 (2005; Zbl 1077.03009)], to mention some of them. These representations usually follow the same strategy used to represent modal algebras (see [G. Sambin and V. Vaccaro, Ann. Pure Appl. Logic 37, No. 3, 249–296 (1988; Zbl 0643.03014)]). That is, given a modal algebra \(A\), first consider its reduct obtained by dropping its modal operator, then use some topological representation for this reduct (Stone duality, Priestley duality, etc.), then enrich the space that represents the reduct of \(A\) with a relational structure to represent the modal operator.

The results of this paper are also based on a duality for the reduct without the modal operator. The main difference lies in the fact that, instead of considering the class of modal algebras determined by the properties of its operations, the author’s approach is to consider the class determined in a specific way to construct of modal operators on powers of a semi-primal algebra given a Kripke frame.

This change of perspective has some advantages. First, this approach allows the author to concentrate on the universal properties of the generating algebra that ensure the existence of a duality, rather than on the particularities of the language. Secondly, the fact of having a finite semi-primal algebra as starting point of the process opens access to the Semi-primal Strong Natural Duality Theorem 3.3.14 in [D. M. Clark and B. A. Davey, Natural dualities for the working algebraist. Cambridge: Cambridge University Press (1998; Zbl 0910.08001)], which is the natural duality on which the author bases his construction.

Several authors have tackled the problem of adding modalities to some classes of algebras with lattice reduct and to obtain topological representations for these classes, e.g., R. Cignoli, S. Lafalce and A. Petrovich [Order 8, No. 3, 299–315 (1991; Zbl 0754.06006)], A. Petrovich [Order 16, No. 1, 1–17 (1999; Zbl 0952.06016)], S. Celani and R. Jansana [Log. J. IGPL 7, No. 6, 683–715 (1999; Zbl 0948.03013)], C. B. Wegener [Stud. Log. 70, No. 3, 339–352 (2002; Zbl 1007.06010)], M. Gehrke, H. Nagahashi and Y. Venema [Ann. Pure Appl. Logic 131, No. 1–3, 65–102 (2005; Zbl 1077.03009)], to mention some of them. These representations usually follow the same strategy used to represent modal algebras (see [G. Sambin and V. Vaccaro, Ann. Pure Appl. Logic 37, No. 3, 249–296 (1988; Zbl 0643.03014)]). That is, given a modal algebra \(A\), first consider its reduct obtained by dropping its modal operator, then use some topological representation for this reduct (Stone duality, Priestley duality, etc.), then enrich the space that represents the reduct of \(A\) with a relational structure to represent the modal operator.

The results of this paper are also based on a duality for the reduct without the modal operator. The main difference lies in the fact that, instead of considering the class of modal algebras determined by the properties of its operations, the author’s approach is to consider the class determined in a specific way to construct of modal operators on powers of a semi-primal algebra given a Kripke frame.

This change of perspective has some advantages. First, this approach allows the author to concentrate on the universal properties of the generating algebra that ensure the existence of a duality, rather than on the particularities of the language. Secondly, the fact of having a finite semi-primal algebra as starting point of the process opens access to the Semi-primal Strong Natural Duality Theorem 3.3.14 in [D. M. Clark and B. A. Davey, Natural dualities for the working algebraist. Cambridge: Cambridge University Press (1998; Zbl 0910.08001)], which is the natural duality on which the author bases his construction.

Reviewer: Leonardo Manuel Cabrer (Firenze)

##### MSC:

08C20 | Natural dualities for classes of algebras |

03G25 | Other algebras related to logic |

06D50 | Lattices and duality |

18B99 | Special categories |

##### Keywords:

modal lattice; modal operator; natural duality; coalgebraic duality; topological duality; semi-primal algebra; Kripke frames; enriched Stone space
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\textit{Y. Maruyama}, J. Pure Appl. Algebra 216, No. 3, 565--580 (2012; Zbl 1263.08002)

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