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In \cite{DvZa3}, we started the investigation of pseudo MV-algebras withsquare roots. In the present paper, we continue to study the structure ofpseudo MV-algebras with square roots focusing on their new characterizations.The paper is divided into two parts. In the present first part, we investigatethe relationship between a pseudo MV-algebra with square root and itscorresponding unital $\ell$-group in the scene of two-divisibility.  In the second part, we find some conditions under which a particular class ofpseudo MV-algebras can be embedded into pseudo MV-algebras with square roots.We introduce and investigate the concepts of a strict square root of a pseudoMV-algebra and a square root closure, and we compare both notions. We show thateach MV-algebra has a square root closure. Finally, using the square root ofindividual elements of a pseudo MV-algebra, we find the greatest subalgebra ofa special pseudo MV-algebra with weak square root.

Representation and Embedding of Pseudo MV-algebras with Square Roots I. Strict Square Roots