Models for the common knowledge logic

In this paper, we discuss models of the common knowledge logic. The common knowledge logic is a multi-modal logic that includes the modal operators $\mathsf{K}_{i}$ ($i\in\mathcal{I}$, where $\mathcal{I}$ is a finite set of agents) and $\mathsf{C}$ in the language. The intended meanings of $\mathsf{K}_{i}\phi$ ($i\in\mathcal{I}$) and $\mathsf{C}\phi$ are ''the agent $i$ knows $\phi$'' ($i\in\mathcal{I}$) and ''$\phi$ is common knowledge among $\mathcal{I}$'', respectively. Semantically, this can be expressed as follows: $\mathsf{C}\phi$ is true if and only if all of $\phi$, $\mathsf{E}\phi$, $\mathsf{E}^{2}\phi$, $\mathsf{E}^{3}\phi,\ldots$ are true, where $\mathsf{E}\phi=\bigwedge_{i\in\mathcal{I}}\mathsf{K}_{i}\phi$. A Kripke frame that satisfies the condition is $\langle W,R_{\mathsf{K}_{i}} (i\in\mathcal{I}), R_{\mathsf{C}}\rangle$, where $R_{\mathsf{C}}$ is the reflexive and transitive closure of $R_{\mathsf{E}}=\bigcup_{i\in\mathcal{I}}R_{\mathsf{K}_{i}}$. We refer to such Kripke frames as CKL-frames. An algebra that satisfies the condition is a modal algebra with modal operators $\mathrm{K}_{i}$ ($i\in\mathcal{I}$) and $\mathrm{C}$, which satisfies that $\mathrm{C}x\leq x$, $\mathrm{C} x\leq\mathrm{E}\mathrm{C} x$, and $\mathrm{C} x$ is the greatest lower bound of the set $\{\mathrm{E}^{n} x\mid n\in\omega\}$, where $\mathrm{E} x=\bigwedge_{i\in\mathcal{I}} \mathrm{K}_{i} x$. We refer to such modal algebras as CKL-algebras. In this paper, we show that the class of CKL-frames is modally definable, whereas the class of CKL-algebras is not. That is, the class of CKL-algebras does not form a variety, and there exists a modal algebra in which the common knowledge logic is valid, but $\mathrm{C}x$ is not the greatest lower bound of the set $\{\mathrm{E}^{n} x\mid n\in\omega\}$.