. Unary Algebras A unary algebra is a set of objects with a single unary operation. The operation is modelled as a mapping from the set to the set. As far as I can tell it was introduced in Birkhoff and Bartee's "Modern Applied Mathematics" as an introduction to the "binary" algebras based on a set and a binary operation: semigroups, monoids, and groups. Partial unary algebras are needed to model finite ordinal data types. PARTIAL_UNARY_ALGEBRA::=Net{ Set::Sets, op::Set<>->Set. finite::@ = Set in FiniteSets. trivial::@ = Set={}. nontrivial::@ = some Set. terminal::=Set~ddef(op), -- set of elements that the operator can not operate on. }=::PARTIAL_UNARY_ALGEBRA. UNARY_ALGEBRA::=Net{ Set::Sets, f::Set>->Set. finite::@ = Set in FiniteSets. trivial::@ = Set={}. nontrivial::@ = some Set. fixedpoints::={ x:Set || f(x)=x }, symmetry(f) ::@(Set---Set)={ a:Set---Set || f o a =a o f }. ()|- (eternal_recurrence_1): For all a:Set, i:0.., j:i+1..( if f^i(a)=f^j(a) then for all k:0.., f^(i+k)(a)=f^(j+k)(a)). ()|- (eternal_recurrence_2): For all a:Set, i:0.., j:i+1..( if a.f^i=a.f^j then for all p:0.., a.f^i=a.f^(i+p*(j-i)) ). These are proved under more restrictive conditions as a result in the theory of relations in Gries and Schneider's "A Logical Approach to Discrete Mathematics". }=::UNARY_ALGEBRA. Unary::=\$ \$UNARY_ALGEBRA. unary::=Unary.Set, For X, unary(X) ::=Unary(Set=>X).Set. Note. The above definitions follow conventions suggested in .See http://www/dick/maths/notn_6_Algebra.html Dynamic_System::=following, .Net |- UNARY_ALGEBRA. For Y:Sets, Invariant::@(Set->Y)={g||g o f = g }, For Y:Sets, Invariant::@(Y->Set)={g||for all y:Y, some y':Y(f(g(y))=g(y')) }, Invariant::@@Set={I:@S||f(I)==>I}. ()|- fixed_points = {x:Set || {x0}in Invariants}. ()|- for A,B:Invariants, A|B in Invariants. For N:@Nat0, f^N=map[x]{x.(f^n) || n:N}, ()|-for x0 in fixed_points, N:@Nat~{}, f^N(x0)=x0. cyclic::={x:Set || for some p (f^(p*Nat0)={x} }. equifinal::={(x,y)||f^Nat0(x)&f^Nat0(y)}, ()|-equifinal in EquivalenceRelations(Set), Basins::= Set/equifinal. ...See W Ross Ashby's "Introduction to Cybernetics"[RossAshby56]. .Hole .Close.Net Dynamic_system For Set:Sets, a:Set---Set, symetrical(a)={f:Set->Set||f o a =a o f}. Convergence::=Net{ |- UNARY_ALGEBRA. |- Topology(Set). LimitSets::@@Set={X:@S || for all N:open, if X==>N the for abf i(f^i(X)==>N)). .Hole ... }=::Convergence. . Unary Morphisms For (S,a),(T,b):Unary, arrow:{->,>--,>==,---,==>,-->,...}, (Unary)(S,a)->(T,b) ::= {f:S arrow T|| f o a = b o f} Unary_congruence(Set,f) ::= {E:equivalence_relation||E ==>f;E;/f}. SP_partition(Set,f) ::= {Set/E||E in unary_congruence(Set,f)}. Note - can be generalised to a machine. .See Also - Automata and Systems Theory