On r→-Sheffer strokes: A new class of directionally monotone functions

Recently, Baczyński et al. introduced the axiomatic definition of fuzzy Sheffer strokes and incorporated the Sheffer stroke operation into the fuzzy logic framework [Fuzzy Sets Syst. 431 (2022) 110-128]. In this paper, we introduce a new class of directionally monotone functions, called $\overrightarrow{r}$-Sheffer strokes. Firstly, we propose the notion of $\overrightarrow{r}$-Sheffer strokes by relaxing the monotonicity of fuzzy Sheffer strokes to the directional monotonicity. And then, we discuss some vital properties of such functions as well as its relationship between fuzzy Sheffer strokes. Subsequently, we give a representation of $\overrightarrow{r}$-Sheffer strokes by means of $\overrightarrow{r}$-pre-conjunctions and fuzzy negations. Meanwhile, we give a characterization of $\overrightarrow{r}$-(constant) Sheffer strokes. Besides, we provide several construction methods of $\overrightarrow{r}$-Sheffer strokes. Interestingly, we show that $\overrightarrow{r}$-pre-conjunctions, $\overrightarrow{r}$-pre-disjunctions, (light) $\overrightarrow{r}$-pre-t-norms, (light) $\overrightarrow{r}$-pre-t-conorms, $\overrightarrow{r}$-(quasi-)overlap and grouping functions, and $\overrightarrow{r}$-implication functions can be obtained through adequate combinations of $\overrightarrow{r}$-Sheffer strokes. Finally, we present an example of a potential application of $\overrightarrow{r}$-Sheffer strokes in fire detectors.