Geodesic
Equational bases for weak monounary varieties
Discussiones Mathematicae. General Algebra and Applications, Tome 22 (2002) no. 1, pp. 87-100.

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It is well-known that every monounary variety of total algebras has one-element equational basis (see [5]). In my paper I prove that every monounary weak variety has at most 3-element equational basis. I give an example of monounary weak variety having 3-element equational basis, which has no 2-element equational basis.
Keywords: partial algebra, weak equation, weak variety, regular equation, regular weak equational theory, monounary algebras 
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Bińczak, Grzegorz. Equational bases for weak monounary varieties. Discussiones Mathematicae. General Algebra and Applications, Tome 22 (2002) no. 1, pp. 87-100. http://geodesic.mathdoc.fr/item/DMGAA_2002_22_1_a6/

[1] G. Bińczak, A characterization theorem for weak varieties, Algebra Universalis 45 (2001), 53-62.

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[4] H. Höft, Weak and strong equations in partial algebras, Algebra Universalis 3 (1973), 203-215.

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[6] L. Rudak, A completness theorem for weak equational logic, Algebra Universalis 16 (1983), 331-337.

[7] L. Rudak, Algebraic characterization of conflict-free varieties of partial algebras, Algebra Universalis 30 (1993), 89-100.