TITLE: Do groups, rings and fields have practical applications in CS? If so, what are some?
QUESTION [85 upvotes]: This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be applied (directly or indirectly) or has given me Aha! moments understanding the principles behind the applications. 
Groups, Rings and Fields have always eluded me. I always thought they were useful (instinctively) but failed to see where/how. Are they just theoretical concepts without practical applications? I hope not. So what are their applications, especially in the field of computer science. No matter how arcane/remote their use I still want to know.

REPLY [8 votes]: Semi-rings are used in computational linguistics for weighted automata algorithms. An automaton is a kind of directed graph with annotated edges (or, sometimes, annotated nodes) which is traversed during processing of some input data, and serves to transform the input data (e.g. translate one language into another), annotate the input data (e.g. tag the words in the input with their corresponding part of speech tags), or do some calculation over the input. 
Weights can be added to the edges of automatons, where the weights are elements of a carrier set of a semi-ring, to allow a wider variety of such calculations; for example, they facilitate working with probabilities. Weights can be real numbers, probabilities, distances, strings, feature structures, sets, matrices, or other elements, depending on the chosen semi-ring.
The semi-ring properties, such as closure properties and algebraic possibilities on the semi-ring, are very important for many language-related automata algorithms: finding shortest paths, longest matches, doing transformations on the automaton (such as determinization and minimization) and many others.
Here is a list of some semi-rings that might be used for automata weights, where each semi-ring consists of a carrier set associated with two operations, abstract plus and abstract times with their respective identity elements. When an automaton is traversed, the weights along each path are multiplied, and then the results of all paths are added to obtain a result.

Boolean semiring: $ \left< \{t,f\}, \vee, \wedge, false, true \right> $
 "A path is $ true $ if all its edges are $ true $; the result is $true$ if there is at least one $true$ path."
real semiring: $ \left< \mathbb{R}, +, \dot{}, 0, 1 \right> $
probabilistic semiring: $ \left< [0,1], +, \dot{}, 0, 1 \right> $
 "Probabilities along each path are multiplied; then the sum of all path probabilities is the result"
tropical semiring: $ \left< \mathbb{R}_\infty^+, min, +, \infty, 0 \right> $
 "If the weights represent edge lengths, the result is the shortest possible path length from start to end node"
Viterby semiring: $ \left< [0,1], max, \dot{}, 0, 1 \right> $
 "maximum probability of any path from start to end node"
log semiring: $ \left< \mathbb{R}_\infty, \oplus_{log}, +, \infty, 0 \right> $ 
with $ x \oplus_{log} y = -ln\left( e^{-x} + e^{-y} \right) $
 "like the probabilistic semiring, but $-log$-transformed to avoid numerical underflow and increase performance; see Intuitive use of logarithms"
arctic semiring: $ \left< \mathbb{R}_{-\infty}, max, +, -\infty, 0 \right> $
 "opposite of tropical; longest paths"
fuzzy semiring: $ \left< [0,1], max, min, 0, 1 \right> $
 "If edges are bridges, weights are maximum load capacities, this will obtain the maximum weight that can be hauled from start node to end node: get strongest path, where each path is only as strong as its weakest link"
concatenation semiring: $ \left< 2^{\Sigma^*}, \cup, \dot{}, \emptyset, \{\epsilon\} \right> $ 
 with $ 2^{\Sigma^*} = $ powerset of all strings $ = $ set of all formal languages,
 $ \epsilon = $ empty string
string semiring: $ \left< \Sigma \cup {s_\infty}, \wedge, \dot{}, s_\infty, \epsilon \right> $ 
 with $s_\infty = $ string which contains all other strings, 
 $ \wedge = $ longest common prefix
set semiring: $ \left< 2^M, \cup, \cap, \emptyset, M \right> $
 where $ M $ is an arbitrary set
unification semiring: $\left< 2^\mathcal{F}, \cup, \sqcup, \emptyset, \{\bot\} \right>  $
 where $\mathcal{F} = $ set of feature structures
 $\sqcup = $ unification
 $\bot = $ bottom, invalid feature structure