DHARMALGEBRAIC STRUCTURES 81Therefore, algebraic system (Σ, .) is a monoid. If we replace the set Σ by Σ+, where Σ+
= Σ* – {ε}, where set Σ+ contains all possible strings formed over the alphabet Σ except null
string (ε) then algebraic system (Σ+, .) is not a monoid due to non existence of an identity
element with respect to operation concatenation but a semigroup.
Operation Table
The properties hold by an algebraic system can be easily observed from the operation table.
Let (X, ) is an algebraic system, where is a binary operation on X, then operation table
presented an arrangement of values resulted after performing the binary operation over
the elements of set X. For example, the algebraic structure (X, ⊕) is a monoid, where ⊕ is a
binary operation defined over set X = {0, 1, 2, 3}, i.e.
a ⊕ b = a + b, if a + b ≤ 3, otherwise a + b – 4
Construct the transition table for algebraic system (X, )
⊕ 0123
0 0123
1 1230
2 2301
3 3012
Fig. 4.2 Operation table.
From the operation table shown in Fig. 4.2 we derive following conclusions,
l Since each element in table belongs to set X, hence operation ⊕ is closed.
l Operation is associative.
l Values of the first column are similar to the corresponding element of the set when
operated on ⊕ over element 0, hence 0 is a unique identity element.
Therefore, algebraic system (X, ⊕) is a monoid. Further, we see that algebraic system
(X, ⊕) is also a group.
l Since there is an occurrence of the identity element 0 in each row in the operation
table which shows the existence of the inverse element for every element of X, i.e.,
0 ⊕ 0 = 0; 1 ⊕ 3 = 0; 2 ⊕ 2 = 0; 3 ⊕ 1 = 0.
l Also corresponding rows and column are same in the table; hence operation ⊕ holds
commutative property.
Therefore, algebraic system (X, ⊕) is an Abelian group.
Example 4.2. Let X = {p, q, r} and binary operation ⊗ defines in the operation table shown in
Fig. 4.3 then algebraic system (X, ⊗) is a semi group but not monoid.
Sol.
⊕ pqr
p ppp
q qqq
r rrr
Fig. 4.3 Operation table.
l Operation ⊗ is closed, since all elements in the table belong to set X.
l Operation is associative.