## Sheffer operation in ortholattices.(English)Zbl 1091.06007

An ortholattice is an algebra $$(L,\vee,\wedge,',0,1)$$ of type $$(2,2,1,0,0)$$ such that $$(L,\vee,\wedge,0,1)$$ is a bounded lattice satisfying $$(x\vee y)'=x'\wedge y'$$, $$(x')'=x$$ and $$x\vee x'=1$$.
An ortho-Sheffer operation on a set $$M$$ is a binary operation on $$M$$ satisfying $$xy=yx$$, $$(xx)(xy)=x$$, $$x((yz)(yz))=((xy)(xy))z$$, $$(x((xx)(yy)))(x((xx)(yy)))=x$$ and $$y(x(xx))=yy$$.
It is proved that for every fixed non-empty set $$L$$ the formulas $$xy=x'\vee y'$$ and $$x\vee y=(xx)(yy)$$, $$x\wedge y=(((xx)(xx))((yy)(yy)))(((xx)(xx))((yy)(yy)))$$, $$x'=xx$$, $$0=(x(xx))(x(xx))$$, $$1=x(xx)$$, respectively, induce mutually inverse bijections between the set of all ortholattices with base set $$L$$ and the set of all ortho-Sheffer operations on $$L$$.

### MSC:

 06C15 Complemented lattices, orthocomplemented lattices and posets 06E30 Boolean functions
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### References:

 [1] Birkhoff G.: Lattice Theory. : Proc. Amer. Math. Soc., Providence, R. I. third edition, 1967. · Zbl 0063.00402 [2] Grätzer G.: General Lattice Theory. : Birkhäuser Verlag, Basel. second edition, 1998. [3] Sheffer H. M.: A set of five independent postulates for Boolean algebras. Trans. Amer. Math. Soc. 14 (1913), 481-488. · JFM 44.0076.01
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