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Max cones are max-algebraic analogs of convex cones. In the present paper we develop a theory of generating sets and extremals of max cones in ${{\mathbb R}}_+^n$. This theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors. We give new proofs of existing results suitably generalizing, restating and refining them. Of these, it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements. We include results on properties of open and closed cones, on properties of totally dependent sets and on computational bounds for the problem of finding the (essentially unique) basis of a finitely generated cone.Comment: 15 pages, 1 figure; v2: new layout, several new references, renumbering of result

By the use of Perron-Frobenius theory, simple proofs are given of the Fundamental Theorem of Demography and of a theorem of Cushing and Yicang on the net reproductive rate occurring in matrix models of population dynamics. The latter result, which is closely related to the Stein-Rosenberg theorem in numerical linear algebra, is further refined with some additional nonnegative matrix theory. When the fertility matrix is scaled by the net reproductive rate, the growth rate of the model is $1$. More generally, we show how to achieve a given growth rate for the model by scaling the fertility matrix. Demographic interpretations of the results are given.

We shall denote the inner product in R" by (z, y) = ZT Y and we write IIzll2 = (z, z), IIzil ~ 0, DEFINITION 2. The polar S* of a nonempty set Sin R n is defined to be S* = {z E R" :(z, y) ~ 0 for all YES}.Since 0 E S*, we observe that S* is non empty. Also it is easily shown that S* is closed.

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