By Jiří Adámek, ing.; Jiří Rosický; E M Vitale

''Algebraic theories, brought as an idea within the Sixties, were a basic step in the direction of a express view of common algebra. in addition, they've got proved very worthwhile in a variety of parts of arithmetic and laptop technology. This conscientiously constructed booklet offers a scientific advent to algebra in line with algebraic theories that's obtainable to either graduate scholars and researchers. it's going to facilitateRead more...

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**Extra info for Algebraic theories : a categorical introduction to general algebra**

**Example text**

Homomorphisms of algebras are represented by natural transformations. We now introduce a more general definition of an algebraic theory and its algebras. See Chapter 11 for Lawvere’s original concept of a one-sorted theory and Chapter 14 for S-sorted theories. In the present chapter, we also study basic concepts such as limits of algebras and representable algebras, and we introduce some of the main examples of algebraic categories. 1 Definition An algebraic theory is a small category T with finite products.

This proves that e is surjective. 18 Corollary Every algebraic category is exact. In fact, since Set is exact, so is Set T . 3. 19 Definition We say that colimits in a category A distribute over products if given diagrams Di : Di → A (i ∈ I ), and forming the diagram Di → A, Ddi = D: i∈I Di di , i∈I the canonical morphism colim D → colim Di i∈I is an isomorphism. If all Di are of a certain type, we say that colimits of that type distribute over products. The concept of distributing over finite products is defined analogously but I is required to be finite.

18. 2. The category of posets does not have this property: take an arbitrary poset B and an equivalence relation R on the underlying set of B equipped with the discrete ordering; the two projections R ⇒ B form an equivalence relation that is seldom a kernel pair. 16 Definition A category is called exact if it has 1. 2. 3. 4. finite limits coequalizers of kernel pairs effective equivalence relations and regular epimorphisms stable under pullback; that is, in every pullback e f A G B C G D g e if e is a regular epimorphism, then so is e .