By van der Waerden B.L.

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If T D fa; bg has two elements, we write a ^ b for the meet (if it exists). 4 A partially ordered set S is a lattice if for any a; b 2 S both a _ b and a ^ b exist. The morphisms in the category of lattices are order-preserving functions that preserve both meets and joins. We can also think of _ and ^ as binary operations on S , and recover the order by defining x Ä y if and only if x ^ y D x. This gives a different way of defining a lattice. 5 Two binary operations _ and ^ on a set S will be the join and meet operators with respect to an order on S if and only if the following identities hold for all a; b; c 2 S : 1.

3 A category D is called a subcategory of C if every object (and arrow) of D is an object (and arrow) of C, the identity arrows in C are also (identity) arrows in D, and composition of arrows in D is the composition inherited from C. 3. Natural Transformations Given a subcategory D, sending each object and arrow to itself is a functor from D to C. This functor is clearly always faithful. A; B/. For example, Ab is a full subcategory of Gr, but the category of rings is not a full subcategory of the category of rngs, since a homomorphism of rngs need not preserve the multiplicative identity element.

1 Let G be a group and X be a set. A left action of G on X is a function G X ! g; x/ 7! h x/ for all g; h 2 G and all x 2 X, and 1 xDx for all x 2 X. It follows that g x D y if and only if x D g 1 y. In particular, for each g 2 G the function x 7! g x must be invertible. If X has extra structure, we usually require that the action respect that structure. For example, if X is a metric space, we might require that the function x 7! g x preserve distances, or, less stringently, that it be continuous.