## 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.

### MSC:

 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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