# The representation of square root quasi-pseudo-MV algebras

Research paper by Wenjuan Chen, Wieslaw A. Dudek

Indexed on: 01 Oct '14Published on: 01 Oct '14Published in: Soft Computing

#### Abstract

$$\sqrt{'}$$ quasi-MV algebras arising from quantum computation are term expansions of quasi-MV algebras. In this paper, we introduce a generalization of $$\sqrt{'}$$ quasi-MV algebras, called square root quasi-pseudo-MV algebras ($$\sqrt{\hbox {quasi-pMV}}$$ algebras, for short). First, we investigate the related properties of $$\sqrt{\hbox {quasi-pMV}}$$ algebras and characterize two special types: Cartesian and flat $$\sqrt{\hbox {quasi-pMV}}$$ algebras. Second, we present two representations of $$\sqrt{\hbox {quasi-pMV}}$$ algebras. Furthermore, we generalize the concepts of PR-groups to non-commutative case and prove that the interval of a non-commutative PR-group with strong order unit is a Cartesian $$\sqrt{\hbox {quasi-pMV}}$$ algebra. Finally, we introduce non-commutative PR-groupoids which extend abelian PR-groupoids and show that the category of negation groupoids with operators and the category of non-commutative PR-groupoids are equivalent.