Posted by: Reetesh Mukul | October 12, 2008

Magma and Monoids – II

Formal Systems will be the atomic system for the analysis of subject matter in this blog, however I have not talked about them in this blog so far. Nevertheless, I have no intention to jump into the intricacies of it at this time, and I assume that the readers of this post have some understanding of it. Let us play with a Formal System, which will be described about the law of composition illustrated in last post. The Magma, E, here is the set \{x_1, x_2, x_3\}. For brevity, the infix symbol for law of composition,  \top, is dropped. Such type of dropping of infix symbols are typically done when law of composition is multiplication; though here we are dropping it for any law of composition.

Symbols: x_1, x_2, x_3

Axioms: (x_1x_1),(x_1x_2),(x_1x_3),(x_2x_1),(x_2x_2),(x_2x_3),(x_3x_1),x_3x_2),(x_3x_3) are compositions.

Rule of Composition: If R and S are composition or symbol, then (RS) is a composition.

Thus in this formal systems we can manufacture following compositions:-

  1. (((x_2x_1)x_2)x_3)
  2. ((((x_2x_1)(x_1x_3))(x_1((x_2x_3)x_1)))((x_1x_3)(x_1x_2)))
  3. (((x_3x_3)(((x_1x_3)(x_3x_3))(x_3x_2))) ((((x_3x_1)x_3)((x_3x_3)(x_2x_1)))  ((x_2x_1)(x_1(x_3(x_1x_1))))))
  4. (((x_2(x_3x_1))(((x_3x_3)(x_1x_1))(x_3x_1)))((x_2x_3)(x_2x_1)))

and many others. But many strings, like x_1x_2x_3, (x_1x_2)x_3x_3; are not  a composition. When law of composition of a given magma is used many a times, the inner pattern of doing composition, is isomorphic to one of the composition strings produced by this Formal system. Consider another Formal System, whose Symbols and Axioms are the same as that of above one, while the Rule of Composition, which we will call Rule of Ordered Sequential Composition (the cause of such title will be clear, soon) has modification:-

Rule of Ordered Sequential Composition: If R is a symbol and S is a composition or symbol, then (RS) is a composition.

Thus we get following compositions:-

  1. (x_1(x_2x_3))
  2. (x_1(x_2(x_1x_3)))
  3. (x_3(x_1(x_2(x_3))))

An observation of patterns of these strings gives a mechanism of creating them:-

  1. Lay some x’s sequentially. For example, as such, x_1 x_2 x_3x_1
  2. Put left parenthesis in between each of the x’s. For above mentioned example as such, x_1(x_2(x_3(x_1
  3. For as many left parenthesis, put the equal number of right parenthesis on extreme left side. For above mentioned example it becomes, x_1(x_2(x_3(x_1)))
  4. Enclose the formed string, in parentheses. Thus for the example, we get, (x_1(x_2(x_3(x_1)))) .

Many people will find above pattern creating procedure, and the Formal systems, too plain, dry;  but then for a pure mechanical system this is the basic requirement for any further action. Also, who knows that Formalizing a system may give deeper unreached perspective? Above mechanism gives a naive thought – under the “Rule of Ordered Sequential Composition”, the compositions created can also be created by an array of x’s; the information about the placement of parentheses is global. A precise set theoretic name for this array is “Ordered Sequence”, which basically is a finite family (x_{\alpha})_{\alpha \in A} ,   A being an indexing set which is totally ordered. The set A maps into set E. And the total ordering A gets because of an increasing bijection of A onto an interval \left[0, n\right] of  N. The Formal System (corresponding to “Rule of Ordered Sequential Composition”) is analogous to Nicolas Bourbaki’s definition for Composition of an ordered sequence of Elements:-

Let (x_{\alpha})_{\alpha \in A} be an ordered sequence of elements in a magma E whose indexing set A is non-empty. The composition( under the law \top) of the ordered sequence (x_{\alpha})_{\alpha \in A} , denoted by  \underset{\alpha \in A}{\top}{x_{\alpha}} is the element of  E defined by induction on the number of elements in A as follows:

  1. if A = \{\beta\} then \underset{\alpha \in A}{\top}{x_{\alpha}} = x_{\beta};
  2. if A has p > 1 elements, \beta is the least element of A and A^{\prime} = A-\{\beta\}, then \underset{\alpha \in A}{\top}{x_{\alpha}} ={x_{\beta}}{\top}{(\underset{\alpha \in A}{\top}{x_{\alpha}})}

The compositions produced by the formal system corresponding to “Rule of Composition” is a superset to that of compositions produced by the formal system corresponding to “Rule of Ordered Sequential Composition”.

In the next post on “Magma and Monoids”, there will be discussion on Associativity and Commutativity. I will end this post with the note that all we have done here are some constructions. A digressive question — Is Mathematics all about constructions ? Constructions are important; constructionism is an ideology, a thought. What is your take?

Posted in Algebra, Algebraic Structures, Mathematics | Tags: , ,

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