Typeclass Law Predicate Equivalence transitivity$a=b ∧ b=c$$⇒$$a=c$ symmetry $a=b$$⇔$$b=a$ reflexivity$a$$=$$a$ Order transitivity$a≤b ∧ b≤c$$⇒$$a≤c$ connectivity$a≤b$$∨$$b≤a$ reflexivity$a$$≤$$a$ complement consistency $a≤b$ $⇒$ $a \ngtr b$ equivalence consistency $a≤b ∧ b≤a$ $⇒$ $a=b$ Bounded lower bounded $a$$≥$ $\minBound$ upper bounded $a$ $≤$ $\maxBound$ Semigroup associativity $(a ⊕ b) ⊕ c$ $=$ $a ⊕ (b ⊕ c)$ combineMany associativity $\combineMany(a, [b, c])$ $=$ $a ⊕ (b ⊕ c)$ Monoid left identity$∅ ⊕ a$$=$$a$ right identity$a ⊕ ∅$$=$$a$ Invariant identity$\imap\left(\id, \id\right)$$=$$\id$ composition $\imap(ab, ba) \∘ \imap(bc, cb)$ $=$ $\imap(bc \∘ ab, ba \∘ cb)$ Contravariant identity$\contramap(\id)$$=$$\id$ composition $\contramap(ba) \∘ \contramap(cb)$ $=$ $\contramap(ba \∘ cb)$ Covariant identity$\map(\id)$$=$$\id$ composition $\map(ab) \∘ \map(bc)$ $=$ $\map(bc \∘ ab)$ Applicative identity$\id \▹ \of \▹ \ap(a)$$=$$a$ homomorphism $ab \▹ \of \▹ (a \▹ \of \▹ \ap)$ $=$ $a \▹ ab \▹ \of$ associative composition $\fbc \▹ \map(\compose) \▹ \ap(\fab) \▹ \ap(\fa)$ $=$ $\fbc \▹ \ap\(\fab \▹ \ap(\fa))$ interchange $\fab \▹ \ap\(\of(a)\)$ $=$ $a \▹ \apply \▹ \of \▹ \ap(\fab)$ map consistency $\fa \▹ \map(ab)$ $=$ $ab \▹ \of \▹ \ap(\fa)$ product consistency $\fab \▹ \ap(\fa)$ $=$ $\product(\fab, \fa) \▹ \map\(([ab, a]) ⇒ ab(a)\)$ flatMap consistency $\ap(\fa)$ $=$ $\flatMap\(ab ⇒ \!\map(\fa, ab)\)$ Monad left identity $\of \∘ \flatMap(\afb)$ $=$ $\afb$ right identity$\flatMap(\of)$$=$$\id$ associativity $\flatMap(\afb) \∘ \flatMap(\bfc)$ $=$ $\flatMap\(\flatMap(\bfc) \∘ \afb\)$ map consistency $\map(ab)$ $=$ $\flatMap(\of \∘ ab)$ SemiAlternative associativity $(\fa_1 ⊕ \fa_2) ⊕ \fa_3$ $=$$\fa_1 ⊕ (\fa_2 ⊕ \fa_3)$ coproductMany associativity $\coproductMany(\fa_1, [\fa_2, \fa_3])$ $=$$\fa_1 ⊕ (\fa_2 ⊕ \fa_3)$ Alternative left identity$∅ ⊕ \fa$$=$$\fa$ right identity $\fa ⊕ ∅$ $=$ $\fa$ coproductAll zero $\coproductAll\:\!\raisebox{0.5pt}{([\,])}$ $=$ $∅$ Traversable identity$\Id.\traverse(\id)$$=$$\id$ composition $\G.\map\(\H.\traverse(bhc)\) \∘ \G.\traverse(agb)$ $=$ $\GH.\traverse\(\G.\map(bhc) \∘ agb\)$ map consistency $\F.\map(ab)$ $=$ $\Id.\traverse(ab)$