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
Full Text:

References:

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