# Monoid: Quiz

Question 1: Indeed, the axioms required of a monoid operation are exactly those required of ________ composition when restricted to the set of all morphisms whose source and target is a given object.
HomomorphismCategory theoryGroup (mathematics)Morphism

Question 2: The set of all functions SS forms a monoid under ________.
Inverse functionExponentiationFunction compositionFunction (mathematics)

Question 3: The transition monoid and syntactic monoid are used in describing finite state machines, whereas trace monoids and history monoids provide a foundation for process calculi and ________.
Functional programmingConcurrent computingProgramming paradigmReflection (computer science)

Question 4: Any ________ S may be turned into a monoid simply by adjoining an element e not in S and defining e*s = s = s*e for all sS.
Group (mathematics)SemigroupAlgebraic structureRing (mathematics)

Question 5: The set of all n by n matrices over a given ring, with matrix addition or ________ as the operation.
Matrix multiplicationField (mathematics)Vector spaceRing (mathematics)

Question 6: Given two sets M and N endowed with monoid structure, their ________ M × N is also a monoid.
Cartesian productDirect productBinary relationCardinal number

Question 7: One does this by extending (finite) ________ on Σ to monoid congruences, and then constructing the quotient monoid, as above.
Order theoryFinitary relationBinary relationEquivalence relation

Question 8: Alternately, the associativity of monoid operations ensures that the operation can be ________ by employing a prefix sum or similar algorithm, in order to utilize multiple cores or processors efficiently.
Vector processorParallel computingSupercomputerGPGPU

Question 9: Fix a monoid M with the operation * and identity element e, and consider its ________ P(M) consisting of all subsets of M.
CardinalityPower setSet (mathematics)Set theory

Question 10: For instance, the result of "folding" a ________ might differ depending on pre-order vs.
B-treeBinary tree (data structure)Binary treeTree (data structure)