Consider the following statements.

A. The identity is unique in any monoid.

B. A monoid is a group if there exists inverse of each element of monoid.

C. Semi group has closure, associative and identity properties.

D. Quasi group has closure property.

Choose the correct answer from the options given below:

Question

Question

This question was previously asked in

UGC NET Computer Science (Paper 2) 17 June 2023 Official Paper

Answer 1

The correct answer is A, B and D only

Key Points

A. The identity is unique in any monoid.
- This is true. There is always exactly one identity element in any monoid structure.
B. A monoid is a group if there exists inverse of each element of monoid.
- This is true. A group must satisfy all the axioms of a monoid and additionally each element must have an inverse.
C. Semi group has closure, associative and identity properties.
- This is false. A semigroup only needs to have closure and associativity properties. The property of identity is not necessary for a structure to be considered a semigroup.
D. Quasi group has closure property.
- This is true. Quasigroups need to have an operation that is defined for every pair of group elements and which outputs an element from the group, providing closure property.