On Permutation Commutative Q-Algebras with Their Ideals

Some of the new algebraic topics investigated in this paper, like permutation commutative Q-algebra, permutation commutative G-part., permutation commutative p-radical., permutation commutative
$\mathbf{p}$
-semisimple and permutation commutative Q-ideal are given and studied. We show that if the associative law is held for any permutation commutative Q-algebra
$(\boldsymbol{X},\#,\boldsymbol{T})$
., then
$X$
is a group under the operation “
$\#\prime\prime$
. Also, the left cancellation law holds in
$G(X)$
., where
$\boldsymbol{G}(\boldsymbol{X})$
is the permutation commutative G-part of
$\boldsymbol{X}$
. Additionally, the homomorphism, kernel, and image of permutation commutative Q-algebras and permutation commutative implicative Q-ideals were outlined with particular outcomes linked to the new concepts that we developed and tested.