Notes on an extension of the structure of frame (Q1201239)

A frame is a complete lattice satisfying the distributivity law \((\bigvee a_ i)\land b=\bigvee(a_ i\land b)\); frame homomorphisms preserve finite meets and all joins. An extended frame (\(E\)-frame) \((A,\cdot)\) is a frame together with a binary operation \(\cdot\) satisfying the rules: (1) \(x\land y\leq x\cdot y\), (2) \((x\land y)\cdot 1=(x\cdot 1)\land (y\cdot 1)\), (3) \((x\land y)\cdot 1=0\Rightarrow x\cdot y=0\), (4) \((x\cdot y)\cdot 1=x\cdot(y\cdot 1)\), (5) \(x\cdot(\bigvee y_ i)=\bigvee(x\cdot y_ i)\), (6) \((\bigvee x_ i)\cdot(y\cdot1)=(\bigvee (x_ i\cdot y))\cdot 1\). An \(E\)-frame homomorphism \(f: (A,\cdot)\to(B,\cdot)\) is a frame homomorphism \(f: A\to B\) satisfying \(f(a)\cdot 1=f(a\cdot 1)\) and \(f(a)\cdot f(b)\leq f(a\cdot b)\). The corresponding categories are denoted by Frm resp. EFrm. If we denote by \(U: \text{EFrm}\to \text{Frm}\) the natural forgetful functor, a natural construction is shown to yield an adjoint for \(U\); a characterization of the extended frames thus obtained is given.