By Turner J., Kautz W.H.

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This method is decomposed into seven phases that can be specialised with the deﬁnition of diﬀerent derivation procedures which sometimes can be used for the derivation of algorithm schemes. 2 Basic Concepts Total Functions. We shall denote the set of total functions from X to Y by [X → Y ] and the formula f ∈ [X → Y ] by f : X → Y . The total functions idX (identity on X), π (ﬁrst projection) and ρ (second projection) are used in this work. The following operations on total functions are employed in this paper: f ◦ g (composition), (f, g) (tupling), dom(f ) (domain), cod(f ) (codomain) and f|X (restriction).

Mn−1 . Suppose that we want to determine whether a product of numbers is divisible by k for some given positive number k . m = m/k where m/k is read as “ m is divisible by k ”. The function dk is the lower adjoint in a Galois connection (dk, kd) between ( Bool , ⇐ ) and ( PosInt , / ) where dk ∈ Bool ← PosInt , kd ∈ PosInt ← Bool and / is the is-divisible-by ordering on positive integers. b . ) The pair algebra corresponding to this Galois connection relates all positive integers to the boolean f alse and the positive integers divisible by k to the boolean true .

This paper is about applying the algebraic properties of logical relations to constructing Galois connections of higherorder type. The paper begins in section 1 with a review of the basic algebraic properties of the arrow operator on relations. Proofs are omitted in this section because most of the results are known. Section 2 contains the main results of the paper. The essential ideas have already been observed by Abramsky [Abr90]; our contribution is to specialise his “uniformisation theorem” to Galois-connected functions.

### A survey of progress in graph theory in the Soviet Union by Turner J., Kautz W.H.

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