bilattice

bi- +‎ lattice

bilattice (plural bilattices)

1. (mathematics, computing) A structure B = (S,⊑₁ ,⊑₂) in which S is a non-empty set, and ⊑₁ and ⊑₂ are partial orderings each giving S the structure of a lattice, determining thus for each of the two lattices the corresponding operations of meet and join.
   • 2015, Janko Bračič, Lina Oliveira, “A characterization of reflexive spaces of operators”, in arXiv‎:

     We show that for a linear space of operators ${\displaystyle {\mathcal {M}}\subseteq {\mathcal {B}}(H_{1},H_{2})}$ the following assertions are equivalent. (i) ${\displaystyle {\mathcal {M}}}$ is reflexive in the sense of Loginov--Shulman. (ii) There exists an order-preserving map ${\displaystyle 1}$ on a bilattice ${\displaystyle Bil({\mathcal {M}})}$ of subspaces determined by ${\displaystyle {\mathcal {M}}}$ , with ${\displaystyle P\leq \psi _{1}(P,Q)}$ and ${\displaystyle Q\leq \psi _{2}(P,Q)}$ , for any pair ${\displaystyle (P,Q)\in Bil({\mathcal {M}})}$ , and such that an operator ${\displaystyle T\in {\mathcal {B}}(H_{1},H_{2})}$ lies in ${\displaystyle {\mathcal {M}}}$ if and only if ${\displaystyle \psi _{2}(P,Q)T\psi _{1}(P,Q)=0}$ for all ${\displaystyle (P,Q)\in Bil({\mathcal {M}})}$ .