This note presents some new results from [1] about the Suszko operator and truth-equational logics, following the works of Czelakowski [11] and Raftery [17]. It is proved that the Suszko operator relative to a truth-equational logic $\mathcal{S}$ preserves suprema and commutes with endomorphisms. Together with injectivity, proved by Raftery in [17], the Suszko operator relative to a truth-equational logic is a structural representation, as defined in [15]. Furthermore, if $\mathsf{Alg}(\mathcal{S})$ is a quasivariety, then the Suszko operator relative to a truth-equational logic is continuous. Finally, it is proved that truth is equationally definable in the class ${\mathsf{LMod}}^{\mathrm{Su}}(\mathcal{S})$ if and only if $\mathsf{Alg}(\mathcal{S})$ is a ${\mathit{\tau }}_{\infty }$-algebraic semantics for $\mathcal{S}$ and the Suszko operator ${\stackrel{\sim }{\mathbf{\Omega }}}_{\mathcal{S}}^{\mathit{Fm}}:\mathcal{T}{h}_{\mathcal{S}}\to {\mathrm{Co}}_{\mathsf{Alg}(\mathcal{S})}\mathit{Fm}$ preserves suprema and commutes with substitutions.

中文翻译：

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与真方程逻辑相关的 Suszko 算子

本笔记根据 Czelakowski [11] 和 Raftery [17] 的工作，介绍了 [1] 中关于 Suszko 算子和真值方程逻辑的一些新结果。证明了相对于真方程逻辑的 Suszko 算子$\mathcal{秒}$保留 suprema 并使用自同态交换。与 Raftery 在 [17] 中证明的注入性一起，Suszko 算子相对于真值方程逻辑是一种结构表示，如 [15] 中所定义。此外，如果$\mathsf{藻类}(\mathcal{秒})$是拟变体，则相对于真方程逻辑的 Suszko 算子是连续的。最后，证明了真理在类中是等式可定义的${\mathsf{模块}}^{苏}(\mathcal{秒})$ 当且仅当 $\mathsf{藻类}(\mathcal{秒})$ 是一个 ${\mathit{\tau }}_{\infty }$- 代数语义 $\mathcal{秒}$ 和 Suszko 算子 ${\stackrel{～}{\mathbf{\Omega }}}_{\mathcal{秒}}^{\mathit{调频}}：\mathcal{吨}{H}_{\mathcal{秒}}\to {\mathrm{公司}}_{\mathsf{藻类}(\mathcal{秒})}\mathit{调频}$ 保留 suprema 并用替换来交换。