In MATHS we document three forms depending on whether the operation has a unit:
Further, whether it has an inverse
Detailed properties are documented in [ math_32_Semigroups.html ]
We also use the same word to refer to the set of elements, when there exists such an operation:
The same word (semigroup) with different capitalization indicates the collection of tpls (Set, op) that form semigroups:
It is also helpful to have the following abreviations:
The above set of definitions is typical of the various ways the name of an
algebraic system can be used. This package defining semigroup and
Semigroup in terms of SEMIGROUP can be quickly introduced by
asserting:
A semigroup is said to be Abelian if its operator commutes.
Abelian semigroups are also called
A lattice is actually two semilattices - where a semilattice is a semigroup with an idempotent operation:
For more on semigroups see [ math_32_Semigroups.html ]
. . . . . . . . . ( end of section One Associative Operator) <<Contents | End>>
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 might be described by Net{radius:Positive Real, center:Point, area:=π*radius^2, ...}.
For a complete listing of pages in this part of my site by topic see [ home.html ]
For a more rigorous description of the standard notations see