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Skew Boolean algebras derived from generalized Boolean algebras. (English) Zbl 1146.06008

A skew lattice is an algebra \((S,\vee,\wedge)\) with associative and idempotent binary operations that satisfy the identities \(x\wedge(x\vee y)=x=x\vee(x\wedge y)\) and \((y\vee x)\wedge x=x=(y\wedge x)\vee x\). Every skew lattice has a partial order given by \(a\geq b\) iff \(a\wedge b=b=b\wedge a\) or, equivalently, \(a\vee b=a=b\vee a\). A skew lattice is called symmetric if \(a\wedge b=b\wedge a\Longleftrightarrow a\vee b=b\vee a\). A Boolean skew lattice is a symmetric skew lattice with zero \((S,\vee,\wedge,0)\) such that for all \(a\in S\), the principal poset ideal \(\lceil a\rceil=\{b\in S\mid a\geq b\}\) is a Boolean sublattice of \(S\). The authors make the set \(\omega(B)=\{(a,b)\in B\times B\mid a\geq b\}\) into a Boolean skew lattice by defining \((a,a')\wedge(b,b')=(a\wedge b,a'\wedge b)\) and \((a,a')\vee(b,b')=(a\vee b,((a'\backslash b)\vee b')\), where \(a\backslash b\) is the relative complement of \(a\wedge b\) in \(\lceil a\rceil\).
Authors’ abstract: “Every skew Boolean algebra \(S\) has a maximal generalized Boolean algebra image given by \(S/{\mathcal D}\), where \(\mathcal D\) is Green’s relation defined initially on semigroups. In this paper we study skew Boolean algebras \(\omega(B)\) constructed from generalized Boolean algebras \(B\) by a twisted product construction for which \(\omega(B)/{\mathcal D}\cong B\). In particular we study the congruence lattice of \(\omega(B)\) with an eye to viewing \(\omega(B)\) as a minimal skew Boolean cover of \(B\). This construction is the object part of a functor \(\omega:\mathbf{GB}\to\mathbf{LSB}\) from the category \(\mathbf{ GB}\) of generalized Boolean algebras to the category \(\mathbf{ LSB}\) of left-handed skew Boolean algebras. Thus we also look at its left adjoint functor \(\Omega:\mathbf{LSB}\to\mathbf{GB}\).”
Reviewer’s remark. The definition of generalized Boolean algebras is missing.

MSC:

06E99 Boolean algebras (Boolean rings)
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