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This study aims at giving some new results about the prime and radical subacts of any act over a semigroup S. The notion of prime radical is used to define a closure operator in the category of S-acts, some categorical properties of which are studied. Moreover, a lattice structure is introduced for the collection of radical subacts of an S-act. Also, two different types of Zariski spaces of prime varieties in the category of acts over a semigroup are presented. It is shown that a certain type of Zariski space of acts is isomorphic to a lattice of radical subacts. [ABSTRACT FROM AUTHOR]

A multiplicative subset of a commutative ring contains the zero element precisely if the set in question meets every prime ideal. While this form of Krull's Lemma takes recourse to transfinite reasoning, it has recently allowed for a crucial reduction to the integral case in Kemper and the third author's novel characterization of the valuative dimension. We present a dynamical solution by which transfinite reasoning can be avoided, and illustrate this constructive method with concrete examples. We further give a combinatorial explanation by relating the Zariski lattice to a certain inductively generated class of finite binary trees. In particular, we make explicit the computational content of Krull's Lemma. [ABSTRACT FROM AUTHOR]