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Oxford Quantum Talks Archive

Discrete duality for downset lattices and their residuated operations

Sam van Gool, Radboud University Nijmegen
Flowin'Cat 2010, October 2010, University of Oxford

Downset lattices, also known as doubly algebraic distributive lattices and as completely distributive prime-algebraic lattices, are very special distributive lattices which are in a simple duality with posets. However, this duality is very closely related to the topological Stone and Priestley dualities, and with the use of canonical extension, we can in many cases work with downset lattices without loss of generality. Additional operations, instead of being merely join- or meet-preserving, are residuated in this setting and correspond dually to relations of one arity higher than the operation. We focus on operations coming from one and the same relation on the dual, or operations coming from related relations on the dual, and explore their correspondence theory. This setting occurs in many applications, among which are modal logics, temporal logics, logics focusing on implication, such as intuitionistic and substructural logics, and more recently in semantics for automata theory (Gehrke, Gregorieff, Pin 2010) and in dynamic epistemic logic (Panangaden, Sadrzadeh 2010).

[video] [streaming video] [slides]