Abstract
In this paper, we introduce the concept of several types of groupoids related to semigroups, viz., twisted semigroups for which twisted versions of the associative law hold. Thus, if \documentclass[12pt]{minimal}
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\begin{document}$$(X,*)$$\end{document}(X,∗) is a groupoid and if \documentclass[12pt]{minimal}
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\begin{document}$$\varphi : X^2\rightarrow X^2$$\end{document}φ:X2→X2 is a function \documentclass[12pt]{minimal}
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\begin{document}$$\varphi (a, b) = (u, v)$$\end{document}φ(a,b)=(u,v), then \documentclass[12pt]{minimal}
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\begin{document}$$(X,*)$$\end{document}(X,∗) is a left-twisted semigroup with respect to \documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}φ if for all \documentclass[12pt]{minimal}
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\begin{document}$$a, b, c\in X$$\end{document}a,b,c∈X, \documentclass[12pt]{minimal}
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\begin{document}$$a*(b*c) = (u*v)*c$$\end{document}a∗(b∗c)=(u∗v)∗c. Other types are right-twisted, middle-twisted and their duals, a dual left-twisted semigroup obeying the rule \documentclass[12pt]{minimal}
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\begin{document}$$(a*b)*c = u*(v*c)$$\end{document}(a∗b)∗c=u∗(v∗c) for all \documentclass[12pt]{minimal}
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\begin{document}$$a, b, c\in X$$\end{document}a,b,c∈X. Besides a number of examples and a discussion of homomorphisms, a class of groupoids of interest is the class of groupoids defined over a field \documentclass[12pt]{minimal}
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\begin{document}$$(X,+, \cdot )$$\end{document}(X,+,·) via a formula \documentclass[12pt]{minimal}
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\begin{document}$$x*y=\lambda x + \mu y$$\end{document}x∗y=λx+μy, with \documentclass[12pt]{minimal}
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\begin{document}$$\lambda , \mu \in X$$\end{document}λ,μ∈X, fixed structure constants. Properties of these groupoids as twisted semigroups are discussed with several results of interest obtained, e.g., that in this setting simultaneous left-twistedness and right-twistedness of \documentclass[12pt]{minimal}
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\begin{document}$$(X,*)$$\end{document}(X,∗) implies the fact that \documentclass[12pt]{minimal}
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\begin{document}$$(X,*)$$\end{document}(X,∗) is a semigroup.
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