Abstract
A nearlattice A is a meet semilattice in which every initial segment is a lattice; A is said to be Boolean if all of these lattices are Boolean. Overriding on A is a binary operation \(\triangleleft\) defined by \( a \triangleleft {b} := \sup\{x\colon (x \le a \mbox{ and } x {\mathrel{\mbox{\,\raisebox{.7ex}{$\scriptstyle|$}\hspace{-.75ex}\raisebox{-.35ex}{$\circ$}}}}b) \mbox{ or } x \le b\} \), where \(x{\mathrel{\mbox{\,\raisebox{.7ex}{$\scriptstyle|$}\hspace{-.75ex}\raisebox{-.35ex}{$\circ$}}}}b\) means that the join of x and b exists. In particular, \({\triangleleft}\) extends the partial join operation. Natural examples of overriding nearlattices are provided by trees and by appropriate algebras of partial functions. In the paper, elementary properties of overriding are studied, an axiomatic description of it is obtained, and several results concerning associativity of the operation and the structure of function nearlattices with overriding are presented. The main result is a representation theorem (in terms of algebras of partial functions) for Boolean nearlattices with associative overriding.
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Cīrulis, J. Nearlattices with an Overriding Operation. Order 28, 33–51 (2011). https://doi.org/10.1007/s11083-010-9152-6
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DOI: https://doi.org/10.1007/s11083-010-9152-6