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By J. K. Truss

ISBN-10: 0198533756

ISBN-13: 9780198533757

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The relation R is undiscrete; 2. the relation R is reflexive, difunctional and right punctual; 3. the relation R is reflexive, difunctional and left punctual; 4. R is a right punctual equivalence relation; 5. R is a left punctual equivalence relation. 2. 15, we know that R is an equivalence relation precisely when it is reflexive and difunctional. Moreover, by symmetry, the right punctual case will imply the left punctual one. So it suffices to prove the equivalence of conditions 1 and 2. Of course, condition 1 implies condition 2.

7, we consider a pointed set (X, 1) provided with a right punctual, difunctional relation R, (1, 1) . For all x, y ∈ X we have 1Rx by right punctuality of R, yRy by reflexivity of R and 1Ry by right punctuality again. This implies yRx, by difunctionality of R. We shall also need the following observation, on the reflexive relation S constructed above from an arbitrary relation R. 12 Let E be a pointed category with finite limits. Given a relation r: R X × Y , the following pullback yields a reflexive relation S on R qqqq qqq qqqq qqqq qqq qqqq qqq qqq q qqqq S qqqq qqqq qqqq qqqq qqqq qqqq r σ qqqq qqq qqqqq R×R R qqqq qqqq qqqq d0 × d1 qqq qqqq qq qqqq X×Y and the following conditions hold: 1.

The full subcategory ComM(E) of commutative objects is stable in E under finite limits and is closed under subobjects. 5. 2. 13 Let E be a unital category. The full subcategory ComM(E) of commutative objects is unital and in this subcategory, each diagram of the form X lX qqqq qqqqq qqqq X ×Y qqq qqqqqq qq rY Y is a coproduct of X and Y . Proof The zero object of E is trivially commutative, thus ComM(E) is pointed. 12 it has finite limits – thus in particular finite products – computed as in E.

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Foundations of mathematical analysis MCet by J. K. Truss

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