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Ring-like structures corresponding to pseudo MV-algebras. (English) Zbl 1165.06005

The aim of the paper is to generalize the natural bijective correspondence between MV-algebras and pseudorings to the case of pseudo-MV algebras. Ring-like structures corresponding to MV-algebras were recently introduced by Chajda and Länger. Pseudo-MV algebras were introduced by Georgescu and Iorgulescu and independently by Rachůnek as generalization of MV-algebras. A noncommutative pseudoring is an algebra \((R,+,\cdot,1)\) of type \((2,2,0)\) such that the following axioms are satisfied:
(NP1) \((xy)z=x(yz)\),
(NP2) \(x1=1x=x\),
(NP3) \((1+x)+1=1+(x+1)=x\),
(NP4) \(x0=0x=0\),
(NP5) \(1+(y+1)(x+1)=(1+y)(1+x)+1\),
(NP6) \((1+(x+1)y)(1+x)=(1+(y+1)x)(1+y)=(1+y)(1+x(1+y))=(1+x)(1+y(1+x))\),
(NP7) \(x(1+(y+1)x)=(y(1+x)+1)y\),
(NP8) \(1+(x(1+y)+1)((x+1)y+1)=x+y\),
where \(0\) denotes the element \(1+1\). The main theorem of the paper gives formulas which induce a mutually inverse bijection between the set of all pseudo-MV algebras and the set of all noncommutative pseudorings. Additionally, the author defines the notion of a strong pseudo-De Morgan algebra and gives formulas which induce a mutually inverse bijection between the set of all strong pseudo-De Morgan algebras and the set of all pseudo-MV algebras.
The results of the paper are very interesting and can be a starting point for future studies.

MSC:

06D35 MV-algebras
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