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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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