On a construction of modular \(\mathrm{GMS}\)-algebras (Q2817416)

A modular GMS-algebra is a universal algebra \((L; \vee,\wedge, ^\circ, 0, 1)\) where \((L;\vee,\wedge, 0, 1)\) is a bounded modular lattice and the unary operation \(^\circ\) satisfies the identities \(x\leq x^{\circ\circ}\), \((x\wedge y)^\circ=x^\circ\vee y^\circ\), \(1^\circ=0\). A modular GMS-algebra \(L\) is called a \(K_2\)-algebra if \(L^{\circ\circ}=\{x\in L\mid x^{\circ\circ}=x\}\) is a distributive lattice and \(L\) satisfies the identities \(x\wedge x^\circ=x^{\circ\circ}\wedge x^\circ\) and \(x\wedge x^\circ\leq y\vee y^\circ\). So called \(K_2\)-quadruples are defined and it is proved that there is a one-to-one correspondence between \(K_2\)-algebras and \(K_2\)-quadruples. Theorem 5.2 shows that two \(K_2\)-algebras are isomorphic iff their associated \(K_2\)-quadruples are isomorphic.