Hoops equipped with square roots

Abstract This paper investigates the existence and structure of square roots in hoops, with a particular focus on cancellative, Wajsberg, and basic hoops that are not necessarily bounded. We introduce novel characterizations and decomposition theorems in this context. Specifically, we show that a cancellative hoop $$\textbf{N}(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , arising from the negative cone of an Abelian $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> -group $$\textbf{G}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , admits a square root $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> if and only if $$\textbf{G}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> is two-divisible. For Wajsberg hoops with square roots, we introduce the notion of strong strict square roots and prove that any such hoop admits a unique decomposition into a direct product of a strong strict hoop and an idempotent hoop (which is, equivalently, a generalized Boolean algebra), by employing the framework of a nested family of hoops. We also investigate square roots on ordinal sums of hoops and define the ordinal sum of a family of square roots. This approach enables us to characterize linearly ordered hoops admitting square roots. Finally, we extend these results to basic hoops with square roots and identify generators for certain special subvarieties within this class.