A Solution to a Problem of D. Lau: Complete Classification of Intervals in the Lattice of Partial Boolean Clones

Abstract : The following natural problem, first considered by D. Lau, has been tackled by several authors recently: Let C be a total clone on 2 := {0, 1}. Describe the interval I(C) of all partial clones on 2 whose total component is C. We establish some results in this direction and combine them with previous ones to show the following dichotomy result: For every total clone C on 2, the set I(C) is either finite or of continuum cardinality. 1. Preliminaries Let k ≥ 2 be an integer and let k be a k-element set. Without loss of generality we assume that k := {0,. .. , k − 1}. For a positive integer n, an n-ary partial function on k is a map f : dom (f) → k where dom (f) is a subset of k n , called the domain of f. Let Par (n) (k) denote the set of all n-ary partial functions on k and let Par(k) := n≥1
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Miguel Couceiro, Lucien Haddad, Karsten Schölzel, Tamas Waldhauser. A Solution to a Problem of D. Lau: Complete Classification of Intervals in the Lattice of Partial Boolean Clones. Journal of Multiple-Valued Logic and Soft Computing, Old City Publishing, 2017, 28 (1), pp.47-58. ⟨hal-01183004⟩

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