Solution

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.