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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{p}$</tex> -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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\boldsymbol{X},\#,\boldsymbol{T})$</tex> ., then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> is a group under the operation " <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\#\prime\prime$</tex> . Also, the left cancellation law holds in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G(X)$</tex> ., where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{G}(\boldsymbol{X})$</tex> is the permutation commutative G-part of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{X}$</tex> . 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.