Bosbach states on $L$-graphs

다국어 초록 (Multilingual Abstract)

In this article, we introduce Bosbach states on $L$-graphs. Firstly, the concept of Bosbach states on $L$-graphs is introduced. Then, some propositions of Bosbach states on $L$-graphs are proved. After that, some special Bosbach states on $L$-graphs are studied. In addition, some examples of Bosbach states on special $L$-graphs are provided. Moreover, we obtain that $s$ is a Bosbach state on $L$-graph $G^{\prime}$ can not be characterized by $s$ being a Bosbach state on $L$-graph $G$, where $G^{\prime}$ is the complement of $L$-graph $G$. On this basis, the concept of complement-preserving Bosbach states is proposed. Moreover, we obtain that in linear residuated lattices, $G_1$ and $G_2$ are isomorphic two $L$-graphs where $h$ is a bijection from $G_1$ into $G_2$ and $s$ is a Bosbach state on $L$-graph $G_1$, then $s$ is also a Bosbach state induced by $h$ on $L$-graph $G_2$; based on Example $3.21$, if $G_1$ and $G_2$ are non-isomorphic two $L$-graphs and $s$ is a Bosbach state on $L$-graph $G_1$, $s$ may be also a Bosbach state on $L$-graph $G_2$. Finally, based on Example 3.23 and 3.24, we obtain that in residuated lattices, $G_1$ and $G_2$ are two isomorphic $L$-graphs, then $s$ is a Bosbach state on $L$-graph $G_2 $ can not be characterized by $s$ being a Bosbach state on $L$-graph $G_1$.