Triangular irreducibility of congruences in quasivarieties
The paper under review deals with forms of irreducibility as well as equational definability of relative congruences in quasivarieties. For a natural number \(m\geq 3\), a quasivariety \(\mathbf{Q}\) and \(\mathbf{A}\in \mathbf{Q}\), a \(\mathbf{Q}\)-congruence \(\Phi\) (i.e. \(\mathbf{A}/\Phi\in \mathbf{Q}\)) is said to be \textit{m-triangularily irreducible} in the lattice \(\text{Con}_\mathbf{Q}(\mathbf A)\) if for every sequence \(a_1, \dots , a_m\) of elements of \(\mathbf{A}\) (repetitions allowed), if \(\bigcap_{1\leq i<j\leq m}(\Theta_{\mathbf{Q}}^{\mathbf{A}}(a_i, a_j)+_{\mathbf{Q}}\Phi)=\Phi\) then \( (a_i,a_j)\in \Phi\) for some \(i<j\). Here, \(\Theta^{\mathbf A}_{\mathbf Q}(a_i,a_j)\) are certain \(\mathbf Q\)-congruences. \(\mathbf{A}\in \mathbf{Q}\) is said to be \(m\)-triangularily irreducible in \(\mathbf{Q}\) if \(\mathbf{0}_\mathbf{A}\) is \(m\)-triangularily meet irreducible in the lattice \(\text{Con}_{\mathbf{Q}}(\mathbf{A})\). It is easy to see that \(\mathbf{Q}_{\mathrm{RFSI}}\subseteq \mathbf{Q}_{m\mathrm{-TRI}}\). For \(m>3\), \(\mathbf{A}\in \mathbf{Q}_{m\mathrm{-TRI}}\) iff \(|A|<m\) or \(\mathbf{A}\in \mathbf{Q}_{\mathrm{RFSI}}\). It follows that if \(\mathbf{A}\in \mathbf{Q}\) then \(\mathbf{A}\) is a subdirect product of members of \(\mathbf{Q}_{m\mathrm{-TRI}}\). The author gives various characterization of finitely generated quasi-varieties via \(\mathbf{Q}\)-congruences of the form \(\bigcap_{1\leq i<j\leq m}\Theta_{\mathbf{Q}}^{\mathbf{A}}(a_i, a_j)\). Here is one of them: \(\mathbf{Q}\) is generated by a finite number of algebras of cardinality less than \(m\) iff for every \(\mathbf{A}\in \mathbf{Q}\) and any sequence \(a_1, \dots , a_m\) of elements of \(\mathbf{A}\) (possibly with repetitions), it is the case that \(\bigcap_{1\leq i<j\leq m}\Theta_{\mathbf{Q}}^{\mathbf{A}}(a_i, a_j)=\mathbf{0}_\mathbf{A}\). Let \(m\geq 3\) and \(\Lambda(x_1, \dots x_m, \underline{u})\) a set of equations in variables \(x_1, \dots , x_m\) and possibly some parameters \(\underline{u}\). A quasi-variety \(\mathbf{Q}\) is said to have \textit{equationally definable \(m\)-triangular meets of (relatively) principal congruences} (\(m\)-EDTPM) with respect to \(\Lambda\) if \(\forall \mathbf{ A}\in \mathbf{Q}\) and any \(a_1, \dots , a_m\in\mathbf A\), \(\bigcap_{1\leq i<j\leq m}\Theta_{\mathbf{Q}}^{\mathbf{A}}(a_i, a_j)=\) the \(\mathbf{Q}\)-congruence on \(\mathbf{A}\) generated by the set of pairs \(\langle p^{\mathbf{a}}(a_1, \dots a_m, \underline{e}), q^{\mathbf{a}}(a_1, \dots a_m, \underline{e})\rangle\) with \(\underline{e}\) ranging over sequences of elements of \(\mathbf A\) of the length of \(\underline{u}\) and \(p\approx q\in \Lambda.\) With regard to this concept, the author presents, among other results, the following: {\parindent=8mm \begin{itemize}\item[(i)] \(\mathbf{Q}\) is generated by a finite class of algebras of cardinality less than \(m\) \((\geq 3)\) iff \(\mathbf{Q}\) has \(m\)-EDTPM and \(\bigcap_{1\leq i<j\leq m}\Theta_{\mathbf{Q}}^{\mathbf{F}}(x_i, x_j)=\mathbf{0}_{\mathbf{F}}\) for any sequence \(x_1, \dots , x_m\) of \(m\) different free generators of the free algebra \(\mathbf{F}:=\mathbf{F}_{\mathbf{Q}}(\omega)\). \item[(ii)] If \(\mathbf{Q}\) is as in the previous item, \(\mathbf{Q}\) has \(m\)-EDTPM with respect to the trivial set of equations \(\{x_1\approx x_1, \dots , x_m\approx x_m\}\). \item[(iii)] If \(\mathbf{Q}\) is as in the previous items and in addition has the \textit{weak extension property} (for \(\mathbf{A}\in \mathbf{Q}\) and \(a,b,c,d\in\mathbf A\), \(\Theta(a,b)\cap \Theta(c,d)=\mathbf{0}_{\mathbf{A}}\Longrightarrow \Theta_{\mathbf{Q}}(a,b)\cap \Theta_{\mathbf{Q}}(c,d)= \mathbf{0}_{\mathbf{A}}\)) then, in order to show that \(\mathbf{Q}\) is finitely based, it is necessary and sufficient to find a finite set of equations and a finitely based quasi-variety containing \(\mathbf{Q}\) such that both quasivarieties have \(m\)-EDTPM with respect to that set of equations. \end{itemize}}
\(m\)-triangularily meet-irreducible relative congruence
equationally definable \(m\)-triangular meets of (relatively) principal congruences
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