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regular variety; subregular variety; deductive system; congruence class; difference system
An algebra ${\mathcal A}= (A,F)$ is subregular alias regular with respect to a unary term function $g$ if for each $\Theta , \Phi \in \text{Con}\,{\mathcal A}$ we have $\Theta = \Phi $ whenever $[g(a)]_{\Theta } = [g(a)]_{\Phi }$ 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 $g(a)$ if and only if $C$ is this generalized deductive system. This method is efficient (needs a finite number of steps).
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