A note on $\lambda$-domains and $\Delta$-domains

Let $R$ be an integral domain. Then $R$ is said to be a $\lambda$-domain if the set of all overrings of $R$ is linearly ordered by inclusion. If $R_1 + R_2$ is an overring of $R$ for each pair of overrings $R_1, R_2$ of $R$, then $R$ is said to be a $\Delta$-domain. We show that if $R\subset T$ is an extension of integral domains such that each proper subring of $T$ containing $R$ is a $\lambda$-domain (resp., $\Delta$-domain), then $T$ is a $\lambda$-domain (resp., $\Delta$-domain under some conditions). Moreover, the pair $(R, T)$ is a residually algebraic pair. Two new ring theoretic properties, namely $\lambda$-property of domains and $\Delta$-property of domains are introduced and studied.