- Unary Algebras
- Unary Morphisms
- Notes on MATHS Notation
- Notes on the Underlying Logic of MATHS
- Glossary

- 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 }.
- (above)|- (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)).
- (above)|- (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 [ 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}.
- (above)|-fixed_points = {x:Set || {x0}in Invariants}.
- (above)|-for A,B:Invariants, A|B in Invariants.
- For N:@Nat0, f^N=map[x]{x.(f^n) || n:N},
- (above)|-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)},
- (above)|-equifinal in EquivalenceRelations(Set),
- Basins::= Set/equifinal.
- ...See W Ross Ashby's "Introduction to Cybernetics"[RossAshby56]. [click here if you can fill this hole]

(End of 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)). [click here if you can fill this hole] ...

}=::Convergence.# Unary Morphisms

- |-UNARY_ALGEBRA.
- 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.
[ Also - Automata and Systems Theory ]

# Notes on MATHS Notation

Special characters are defined in [ intro_characters.html ] that also outlines the syntax of expressions and a document.Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block Structure in logic_25_Proofs ] for more on the structure and rules.

The notation also allows you to create a new network of variables and constraints. A "Net" has a number of variables (including none) and a number of properties (including none) that connect variables. You can give them a name and then reuse them. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents. A quick example: a circle = Net{radius:Positive Real, center:Point}.

For a complete listing of pages in this part of my site by topic see [ home.html ]

# Notes on the Underlying Logic of MATHS

The notation used here is a formal language with syntax and a semantics described using traditional formal logic [ logic_0_Intro.html ] plus sets, functions, relations, and other mathematical extensions.For a more rigorous description of the standard notations see

- STANDARD::= See http://www.csci.csusb.edu/dick/maths/math_11_STANDARD.html
# Glossary

- above::reason="I'm too lazy to work out which of the above statements I need here", often the last 3 or 4 statements. The previous and previous but one statments are shown as (-1) and (-2).
- given::reason="I've been told that...", used to describe a problem.
- given::variable="I'll be given a value or object like this...", used to describe a problem.
- goal::theorem="The result I'm trying to prove right now".
- goal::variable="The value or object I'm trying to find or construct".
- let::reason="For the sake of argument let...", introduces a temporary hypothesis that survives until the end of the surrounding "Let...Close.Let" block or Case.
- hyp::reason="I assumed this in my last Let/Case/Po/...".
- QED::conclusion="Quite Easily Done" or "Quod Erat Demonstrandum", indicates that you have proved what you wanted to prove.
- QEF::conclusion="Quite Easily Faked", -- indicate that you have proved that the object you constructed fitted the goal you were given.
- RAA::conclusion="Reducto Ad Absurdum". This allows you to discard the last
assumption (let) that you introduced.

Partial unary algebras are needed to model finite ordinal data types.