Quasi group
A quasi group is a set equipped with a binary operation that satisfies the Latin square property, meaning that for any two elements, there exists unique solutions to the equations of the form `ax=b` and `ya=b`. In the context of square matrices of order 2 with respect to matrix multiplication, the set satisfies this property.

Semi group
A semi group is a set with an associative binary operation. In the case of square matrices of order 2 with respect to matrix multiplication, this set forms a semi group as matrix multiplication is associative. However, it does not necessarily have an identity element or inverses for every element, which are required for a monoid or a group.

Monoid
A monoid is a semi group with an identity element. In the case of square matrices of order 2 with respect to matrix multiplication, this set does not have an identity element as not every 2x2 matrix has a multiplicative identity.

Group
A group is a monoid in which every element has an inverse. In the context of square matrices of order 2 with respect to matrix multiplication, this set does not form a group as not every 2x2 matrix has an inverse. For example, singular matrices do not have an inverse, so the set of 2x2 matrices does not satisfy the group properties.
In conclusion, the set of square matrices of order 2 with respect to matrix multiplication forms a semi group but does not satisfy the properties of a monoid or a group due to the absence of an identity element and inverses for every element.