Bilattice
noun ·Rare ·Advanced level
Definitions
- 1 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.
"We show that for a linear space of operators #123;#92;mathcalM#125;#92;subseteq#123;#92;mathcalB#125;(H#95;1,H#95;2) the following assertions are equivalent. (i) #123;#92;mathcalM#125; is reflexive in the sense of Loginov--Shulman. (ii) There exists an order-preserving map 1 on a bilattice Bil(#123;#92;mathcalM#125;) of subspaces determined by #123;#92;mathcalM#125;, with P#92;leq#92;psi#95;1(P,Q) and Q#92;leq#92;psi#95;2(P,Q), for any pair (P,Q)#92;inBil(#123;#92;mathcalM#125;), and such that an operator T#92;in#123;#92;mathcalB#125;(H#95;1,H#95;2) lies in #123;#92;mathcalM#125; if and only if #92;psi#95;2(P,Q)T#92;psi#95;1(P,Q)#61;0 for all (P,Q)#92;inBil(#123;#92;mathcalM#125;)."
Example
More examples"We show that for a linear space of operators #123;#92;mathcalM#125;#92;subseteq#123;#92;mathcalB#125;(H#95;1,H#95;2) the following assertions are equivalent. (i) #123;#92;mathcalM#125; is reflexive in the sense of Loginov--Shulman. (ii) There exists an order-preserving map 1 on a bilattice Bil(#123;#92;mathcalM#125;) of subspaces determined by #123;#92;mathcalM#125;, with P#92;leq#92;psi#95;1(P,Q) and Q#92;leq#92;psi#95;2(P,Q), for any pair (P,Q)#92;inBil(#123;#92;mathcalM#125;), and such that an operator T#92;in#123;#92;mathcalB#125;(H#95;1,H#95;2) lies in #123;#92;mathcalM#125; if and only if #92;psi#95;2(P,Q)T#92;psi#95;1(P,Q)#61;0 for all (P,Q)#92;inBil(#123;#92;mathcalM#125;)."
Etymology
From bi- + lattice.
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Data sourced from Wiktionary, WordNet, CMU, and other open linguistic databases. Updated March 2026.