Semiring examples from formal language theory
There are plenty of semirings related to language theory. First of all, the Boolean semiring. Next any class of languages closed under finite union and (concatenation) product is a subsemiring of the semiring of all languages. For instance the rational (= regular) languages form a semiring. See also the related notion of Kleene algebra.

The $k \times k$ matrices over a semiring do form a semiring. In particular, matrices over the Boolean semiring encode nondeterministic finite automata and matrices over the slightly larger semiring $\{-\infty, 0, 1\}$ encode transitions of a Büchi automaton. Matrices over a semiring are used to characterize rational series.

The tropical semirings, in particular $(\mathbb{N} \cup \{+\infty\}, \min, +)$ and $(\mathbb{N} \cup \{-\infty\}, \max, +)$ play a proeminent role in automata theory. They also led to a new branch of mathematics, the tropical geometry.