An algebra $𝒜=(A,F)$ is subregular alias regular with respect to a unary term function $g$ if for each $\Theta ,\Phi \in \text{Con}𝒜$ we have $\Theta =\Phi$ whenever ${\left[g\left(a\right)\right]}_{\Theta }={\left[g\left(a\right)\right]}_{\Phi }$ for each $a\in A$. We borrow the concept of a deductive system from logic to modify it for subregular algebras. Using it we show that a subset $C\subseteq A$ is a class of some congruence on $\Theta$ containing $g\left(a\right)$ if and only if $C$ is this generalized deductive system. This method is efficient (needs a finite number of steps).

It is well known that, given an endofunctor $H$ on a category $ℂ$, the initial $(A+H-)$-algebras (if existing), i.e., the algebras of (wellfounded) $H$-terms over different variable supplies $A$, give rise to a monad with substitution as the extension operation (the free monad induced by the functor $H$). Moss [17] and Aczel, Adámek, Milius and Velebil [2] have shown that a similar monad, which even enjoys the additional special property of having iterations for all guarded substitution rules (complete iterativeness),...

It is well known that, given an endofunctor H on a category C , the initial (A+H-)-algebras (if existing), i.e. , the algebras of (wellfounded) H-terms over different variable supplies A, give rise to a monad with substitution as the extension operation (the free monad induced by the functor H). Moss [17] and Aczel, Adámek, Milius and Velebil [12] have shown that a similar monad, which even enjoys the additional special property of having iterations for all guarded substitution rules (complete...

We show that group conjugation generates a proper subvariety of left distributive idempotent groupoids. This subvariety coincides with the variety generated by all cancellative left distributive groupoids.