Horizontal sums of basic algebras. (English) Zbl 1194.06005

A basic algebra is an algebra \({\mathcal A}=(A,\oplus,\neg,0)\) of type (2,1,0) satisfying the identities \(x\oplus0=x,\neg\neg x=x,\neg(\neg x\oplus y)\oplus y=\neg(\neg y\oplus x)\oplus x\) and \(\neg(\neg(\neg(x\oplus y)\oplus y)\oplus z)\oplus(x\oplus z)=1\). The operations \(x\vee y=\neg(\neg x\oplus y)\oplus y\) and \(x\wedge y=\neg(\neg x\vee\neg y)\) induce a lattice strucure \({\mathcal L}({\mathcal A})=(A,\vee,\wedge,0,1)\) on \(A\). If \({\mathcal L}({\mathcal A})\) is a chain, then \(\mathcal A\) is called a chain basic algebra. Conversely, if \({\mathcal L}=(L,\vee,\wedge,0,1)\) is a bounded lattice where for each \(a\in L\) the mapping \(x\mapsto x^a=\neg x\oplus a\) is an antitone involution on the interval \([a,1]\), then the operations \(x\oplus y=(\neg x\vee y)^a\) and \(\neg x=x^0\) induce a basic algebra structure \({\mathcal A}({\mathcal L})=(L,\oplus,\neg,0)\) on \(L\).
Let \(({\mathcal L}_{\gamma})_{\gamma\in\Gamma}\) be a family of bounded lattices \({\mathcal L}_{\gamma}=(L_{\gamma},\vee,\wedge,0,1)\), where the sets \(L_{\gamma}\backslash\{0,1\}\) are mutually disjoint. Let \(L\) be the set obtained from \(\bigcup_{\gamma\in\Gamma} L_\gamma\) after identification of all 0s and identification of all 1s. The horizontal sum of the lattices \({\mathcal L}_{\gamma}\) is the lattice defined on \(L\) by the operations
\[ x\vee y=x\vee y\;\text{ in }\;L_{\gamma}\;\text{ if }\;x,y\in L_{\gamma},\;\text{ else }\;1, \]
\[ x\wedge y=x\wedge y\;\text{ in }\;L_{\gamma}\;\text{ if }\;x,y\in L_{\gamma},\;\text{ else }\;0. \]
If \(({\mathcal A}_{\gamma})_{\gamma\in\Gamma}\) is a family of basic algebras, then \({\mathcal A}({\mathcal L})\) is a horizontal sum of the algebras \({\mathcal A}_{\gamma}\) if \(\mathcal L\) is a horizontal sum of the lattices \({\mathcal L}({\mathcal A}_{\gamma})\). The present paper provides necessary and sufficient conditions for a basic algebra to be a horizontal sum of chain basic algebras, of MV-algebras and of Boolean algebras, respectively.


06D35 MV-algebras
06E05 Structure theory of Boolean algebras
06C15 Complemented lattices, orthocomplemented lattices and posets
03G25 Other algebras related to logic
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