REDEFINITION: AN ALGEBRA (⇐) WITH WELLDEFINED OPERATIONS FORMS AN OPERATIONAL GROUP FOR EACH WELLDEFINED OPERATION..NEWDEFINITION: A QUALGEBRA ("quasi-algebra") IS A STRUCTURE THAT BECOMES AN ALGEBRA WHEN ANY OPERATIONAL MONOID WITHIN IT ATTAINS AN INVERSE.
To understand this language, and why AN ALGEBRA (⇐) IS MATHEMATICS AT ITS (INCORRIGIBLE) BEST(!), I'll briefly DEFINE and EXPLAIN. We arrive at OPERATION via RELATION and FUNCTION: RELATION → FUNCTION →OPERATION.
RELATIONS Standard form: RELATION(RELATUM 1, RELATUM 2, ..., RELATUM n), that is, relations are "numerically-classified" by NUMBER OF THE RELATA COMPREHENDED WITHIN RELATION.
Thus, in "Bill loves Mary", the RELATION "loves" is a BINARY RELATION, with 2 RELATA, namely, "Bill" and "Mary". In "Bill gives roses to Mary", the RELATION "gives" is TERNARY, with 3 RELATA, namely, "Bill", "roses", "Mary". Etc.
RELATIONS ARE ORDINALLY CLASSIFIED AS
- NULLARY (OR CONSTANTS) -- no RELATA;
- UNARY (as in the Successor Function or a logical predicate)-- single RELATUM;
- BINARY -- 2 RELATA;
- TERNARY -- 3 RELATA;
- QUATERNARY -- 4 RELATA;
- ... ; or, in general, n-ARY RELATIONS with n RELATA.
Set-wise, these are ORDERED n-tuples (as in RELATIONAL DATABASES SUCH AS ORACLE) such that CHANGING ORDER MAY CHANGE THE RELATION.
However, ALL RELATIONS (from 2-ARY on up) can be MODELED BY THE BINARY RELATION. (Thus, the TERNARY RELATION OF "gives" can be written as A BINARY OF BINARIES: gives[roses, [Bill, Mary]], where the ORDERING IN THE 2nd BINARY RELATES GIVER TO GIFTEE.) Hence, we can EXPLAIN ALL n-ARY RELATIONS, for n > 1, by EXPLAINING BINARY RELATIONS.
A BINARY RELATION IS AN ORDERING (ORDERED CORRESPONDENCE) OF TWO MEMBERS OR RELATA, EXISTING IN FOUR FORMS:
- A ONE-MANY RELATION, illustrated by the BINARY RELATION, "his children", where the 1st MEMBER OF THE ORDERING IS "father", the 2ND IS "children": one father can have many children.
- A MANY-ONE RELATION, as in "children of a man";
- A MANY-MANY RELATION, illustrated by "friend". Thus, "Bill is a friend of Mary, but Bill has many other friends and so does Mary (popular girl!)."
- A ONE-ONE RELATION, illustrated by "monogamously married".
The three most important RELATIONS in ARITHMETIC and in many other fields of MATH are THE EQUIVALENCE RELATION ("equals", denoted "=") and THE TWO INEQUIVALENCE RELATIONS ("less than", denoted "<"; and "greater than", denoted ">"). (It's easy to use SETS TO CREATE RELATIONS, and especially the EQUIVALENCE and INEQUIVALENCE RELATIONS.)
FUNCTION: A MANY-ONE OR ONE-ONE RELATION.Thus, "children of a man" (MANY-ONE RELATION) and "monogamously married" (ONE-ONE RELATION) above are also FUNCTIONS; but "his children" (ONE-MANY RELATION) and "friend" (MANY-MANY RELATION), above, are NOT FUNCTIONS.
Please note that A FUNCTION CAN BE THOUGHT OF AS HAVING AN INPUT (DOMAIN OF FUNCTION) AND AN OUTPUT (CODOMAIN OF FUNCTION). This is relevant to what follows.
OPERATION: A ONE-ONE FUNCTION SUCH THAT THE CODOMAIN (OUTPUT) IS A SUBSET OF THE DOMAIN (INPUT).How can that second condition fail? Easy. THE CATALOG FUNCTION LINKS ONE CATALOG NUMBER TO ONE ITEM IN INVENTORY -- so is ONE-ONE -- BUT NUMBERS (DOMAIN ELEMENTS) ARE NOT ITEMS (CODOMAIN ELEMENT), hence, it fails to be a FUNCTION.
DIAGRAM. TAKE A BINARY RELATION TO LUNCH TODAY!
GROUP: A SET CLOSED UNDER AN OPERATION WHICH HAS AN INVERSE.I teach Kids about "The Creeping Baby Group". Any creep (on "all fours") followed by (CONCATENATED WITH) a creep is EQUIVALENT TO A SINGLE CREEP (CLOSURE UNDER CONCATENATION OPERATION). Until Baby learns to CREEP BACKWARD (INVERSE CREEP), Baby can get into a cul-de-sac and yell until retrieved. This is a "MONOID" and a QUALGEBRA. But when Baby learns BACK-CREEP -- INVERSE, SO MONOID BECOMES GROUP, QUALGEBRA BECOMES ALGEBRA -- look out! Baby can go "anywhere"!
An OPERATION INDUCES CLOSURE (CODOMAIN WITHIN DOMAIN). So, INVERSE COMPLETES GROUP.
OK. So what? THE GROUP CONCEPT IS BEHIND EVERY "NATURAL LAW" OF PHYSICS, AND THE MATH HAS LEAD TO FULFILLED PREDICTIONS!
Thus, an ALGEBRA (as I've defined it) HAS ALL WE NEED FOR SURVIVAL AND MORE. Incorrigible?
The Australian philosopher, Douglas Gasking, taught me that "Mathematics is incorrigible. Physics is corrigible."INCORRIGIBLE? When a MATH-MODEL DOESN'T MATCH REALITY, DON'T CORRECT THE MATH -- FIND A DIFFERENT MODEL. But when THE PHYSICS DOESN'T MATCH REALITY, CHANGE THE PHYSICS.
Example: 2 + 2 = 4. But 2 cups of water mixed with 2 cups of alcohol don't quite fill up a 4-cup measure. Why? The smaller alcohol molecules pack "into" the water molecules. Similarly, 2 barrels of tennis balls and 2 barrels of marbles don't fill a 4-barrel structure, because the marbles fill in the interstices between the tennis balls. The failure of this MATH MODEL led to the DISCOVERY OF MOLECULES!