Free \(Q\)-distributive lattices from meet semilattices
A \(Q\)-distributive lattice is an algebra \((L,\wedge ,\vee ,0,1,\nabla)\) of type \((2,2,0,0,1)\) such that \((L,\wedge ,\vee ,0,1)\) is a distributive \((0,1)\)-lattice and \(\nabla\) is a unary operation of \(L\) satisfying the identities \(\nabla 0=0\), \(x\leq\nabla x\), \(\nabla (x\wedge\nabla y)=\nabla (x\wedge y)\), and \(\nabla (x\vee y)=\nabla x\vee\nabla y\). We say that \((L,\wedge ,\vee ,0,1,\nabla)\) is a \(Q\)-distributive lattice freely generated by a poset \(P\), if there exists an injective order preserving mapping \(\phi :P\to L\) such that for every order preserving mapping \(\psi\) from \(P\) into a \(Q\)-distributive lattice \((L',\wedge ,\vee ,0,1,\nabla)\) there exists exactly one homomorphism of \(Q\)-distributive lattices \(f:(L,\wedge ,\vee ,0,1,\nabla)\to (L',\wedge ,\vee ,0, 1,\nabla)\) with \(f\circ\phi =\psi\). For a finite poset \(P\), an explicit construction of a \(Q\)-distributive lattice freely generated by \(P\) is presented. The construction has two steps. The first step constructs the poset of meet-irreducible elements of a freely generated \(Q\)-distributive lattice by two applications of the construction of a free semilattice. The second step uses the construction of the distributive \((0,1)\)-lattice with a given poset of meet-irreducible elements. The construction is illustrated by examples.
freely generated \(Q\)-distributive lattice
scientific article; zbMATH DE number 1533336