Functional representation of finitely generated free algebras in subvarieties of BL-algebras (Q2007879)

The authors provide a functional representation of the finitely generated free algebras in a subvariety \(\mathcal{MS}\) of BL-algebras generated by the ordinal sum \(\mathfrak{S}=[0,1]_{MV}\oplus H\) of the standard MV-algebra \([0,1]_{MV}\) and an arbitrary totally ordered basic hoop \(H\). They show the characterization of the free algebras for the cases of one (Lemma 3.2) and two generators (Theorem 3.19) and moreover a characterization of free algebras in \(n\) generators (Theorem 4.16). Lemma 3.2. \ If \(f\in \mathrm{Free}_{\mathcal{MV}}(1)\) is a function such that \(f(1) = 0\) then the function \[(*) \ \ \ \mathcal{F}(x) = \left\{ \begin{array}{ll} f(x) & (x\in [0, 1]_{MV}\\ 0 & (x\in H) \end{array} \right. \] is in \(\mathrm{Free}_{\mathcal{MS}}(1)\). Conversely, if \(\mathcal{F}\in \mathrm{Free}_{\mathcal{MS}}(1)\) is such that \(F(1) = 0\), then there is a function \(f\in \mathrm{Free}_{\mathcal{MV}}(1)\) which satisfies \((*)\). Theorem 3.19. \ \(\mathcal{F}\in \mathrm{Free}_{\mathcal{MS}}(2)\) if and only if \(\mathcal{F}\) is given by a MS-quadruple \((f_1, f_2, f_3, f_4)\), where for four functions \(f \in \mathrm{Free}_{\mathcal{MS}}(2)\), \(g\in \mathrm{Free}_{\mathcal{H}}(2) \cup \{0\}\) and \(h_x\), \(h_y\), \(f\)-\(x\)-\(H\)-McNaughton and \(f\)-\(y\)-\(H\)-McNaughton functions, respectively, a function \(\mathcal{F} : \mathfrak{S}^2 \to \mathfrak{S}\) is given by a MS-quadruple \((f, h_x, h_y, g)\) if it satisfies: \[ \mathcal{F}(x, y) = \left\{ \begin{array}{ll} f(x, y) & ((x, y)\in [0, 1]^2_{\mathrm{MV}})\\ h_x(x, y) & ((x, y) \in [0, 1]_{\mathrm{MV}} \times H)\\ h_y(x, y) & ((x, y) \in H \times [0, 1]_{\mathrm{MV}}) \\ g(x, y) & ((x, y)\in H \times H ) \end{array} \right. \] whenever \(\mathcal{F}(1, 1) = 1\), or \[ \mathcal{F}(x, y) = \left\{ \begin{array}{ll} f(x, y) & ((x, y)\in [0, 1]^2_{\mathrm{MV}})\\ h_x(x, y) & ((x, y) \in [0, 1]_{\mathrm{MV}} \times H)\\ h_y(x, y) & ((x, y) \in H \times [0, 1]_{\mathrm{MV}}) \\ 0 & ((x, y)\in H \times H ) \end{array} \right. \] whenever \(\mathcal{F}(1, 1) = 0\). Theorem 4.16. \ Let \(\mathcal{F} : \mathfrak{S}^n \to \mathfrak{S}\) be a function given by the \(2^n\)-tuple \((f, \{h_A : \emptyset \neq A \subset \{x_1, \cdots, x_n\} \}, g)\), where \(f\in \mathrm{Free}_{\mathcal{MV}}(n), h_A\) is a \(f\)-\(A\)-\(H\)-McNaughton function and \(g\in \mathrm{Free}_{\mathcal{H}}(n) \cup \{0\}\), i.e., for every \(x\in \mathfrak{S}^n\) the function is given by: \[\mathcal{F}(\bar{x}) = \left\{ \begin{array}{ll} f(\bar{x}) & (\bar{x}\in [0, 1]^n_{\mathrm{MV}} \\ h_A(\bar{x}) & \bar{x} \in R_A\\ g([\bar{x}) & (\bar{x} \in H^n \end{array} \right. \] Then \(\mathcal{F} \in \mathrm{Free}_{\mathcal{MS}}(n)\).