Left Cheban loops are loops that satisfy the identity $x(xy \cdot z) = yx \cdot xz$. Right Cheban loops satisfy the mirror identity $(z \cdot yx)x = zx \cdot xy$. Loops that are both left and right Cheban are called Cheban loops. Cheban loops can also be characterized as those loops that satisfy the identity $x(xy \cdot z) = (y \cdot zx)x$. These loops were introduced by A. M. Cheban. Here we initiate a study of their structural properties. Left Cheban loops are left conjugacy closed. Cheban loops are weak inverse property, power associative, conjugacy closed loops; they are centrally nilpotent of class at most two.
References
A. M. Cheban. Loops with identities of length four and of rank three. II (Russian). In “General Algebra and Discrete Geometry”, pp. 117–120, 164, “Shtiintsa”, Kishinev, Moldova, 1980. MR647654 A. M. Cheban. Loops with identities of length four and of rank three. II (Russian). In “General Algebra and Discrete Geometry”, pp. 117–120, 164, “Shtiintsa”, Kishinev, Moldova, 1980. MR647654
P. Csörgö. Extending the structural homomorphism of LCC loops. Comment. Math. Univ. Carolin., 46 (2005), 385–389. MR2174517 1106.20051 P. Csörgö. Extending the structural homomorphism of LCC loops. Comment. Math. Univ. Carolin., 46 (2005), 385–389. MR2174517 1106.20051
A. Drápal. On multiplication groups of left conjugacy closed loops. Comment. Math. Univ. Carolin., 45 (2004), 223–236. MR2075271 1101.20035 A. Drápal. On multiplication groups of left conjugacy closed loops. Comment. Math. Univ. Carolin., 45 (2004), 223–236. MR2075271 1101.20035
E. G. Goodaire and D. A. Robinson. A class of loops which are isomorphic to all loop isotopes. Canad. J. Math., 34 (1982), 662–672. MR663308 10.4153/CJM-1982-043-2 E. G. Goodaire and D. A. Robinson. A class of loops which are isomorphic to all loop isotopes. Canad. J. Math., 34 (1982), 662–672. MR663308 10.4153/CJM-1982-043-2
M. K. Kinyon and K. Kunen. Power-associative, conjugacy closed loops. J. Algebra, 304 (2006), 679–711. MR2264275 1109.20057 10.1016/j.jalgebra.2006.01.034 M. K. Kinyon and K. Kunen. Power-associative, conjugacy closed loops. J. Algebra, 304 (2006), 679–711. MR2264275 1109.20057 10.1016/j.jalgebra.2006.01.034
M. K. Kinyon and K. Kunen. The structure of extra loops. Quasigroups Related Systems, 12 (2004), 39–60. MR2130578 1076.20065 M. K. Kinyon and K. Kunen. The structure of extra loops. Quasigroups Related Systems, 12 (2004), 39–60. MR2130578 1076.20065
W. W. McCune and R. Padmanabhan. Automated Deduction in Equational Logic and Cubic Curves. Lecture Notes in Computer Science, Vol. 1095. Lecture Notes in Artificial Intelligence. Springer-Verlag, Berlin, 1996. MR1439047 0921.03011 W. W. McCune and R. Padmanabhan. Automated Deduction in Equational Logic and Cubic Curves. Lecture Notes in Computer Science, Vol. 1095. Lecture Notes in Artificial Intelligence. Springer-Verlag, Berlin, 1996. MR1439047 0921.03011
J. M. Osborn. Loops with the weak inverse property. Pacific J. Math., 10 (1960), 295–304. MR111800 0091.02101 euclid.pjm/1103038641
J. M. Osborn. Loops with the weak inverse property. Pacific J. Math., 10 (1960), 295–304. MR111800 0091.02101 euclid.pjm/1103038641
H. O. Pflugfelder. Quasigroups and Loops: Introduction. Sigma Series in Pure Mathematics, Vol. 7. Heldermann Verlag, Berlin, 1990. MR1125767 0715.20043 H. O. Pflugfelder. Quasigroups and Loops: Introduction. Sigma Series in Pure Mathematics, Vol. 7. Heldermann Verlag, Berlin, 1990. MR1125767 0715.20043
J. D. Phillips. A short basis for the variety of WIP PACC-loops. Quasigroups Related Systems, 14 (2006), 73–80. MR2268827 1123.20063 J. D. Phillips. A short basis for the variety of WIP PACC-loops. Quasigroups Related Systems, 14 (2006), 73–80. MR2268827 1123.20063
J. D. Phillips. The Moufang laws, global and local. J. Algebra Appl., 8 (2009), 477–492. MR2555515 1190.20052 10.1142/S021949880900345X J. D. Phillips. The Moufang laws, global and local. J. Algebra Appl., 8 (2009), 477–492. MR2555515 1190.20052 10.1142/S021949880900345X
J. D. Phillips and P. Vojtěchovský. The varieties of loops of Bol-Moufang type. Algebra Universalis, 54 (2005), 259–271. MR2219409 1102.20054 10.1007/s00012-005-1941-1 J. D. Phillips and P. Vojtěchovský. The varieties of loops of Bol-Moufang type. Algebra Universalis, 54 (2005), 259–271. MR2219409 1102.20054 10.1007/s00012-005-1941-1
L. R. Soĭkis. The special loops (Russian). In “Questions of the Theory of Quasigroups and Loops”. pp. 122–131, Redakc.-Izdat. Otdel Akad. Nauk Moldav. SSR, Kishinev, 1970. MR281828 L. R. Soĭkis. The special loops (Russian). In “Questions of the Theory of Quasigroups and Loops”. pp. 122–131, Redakc.-Izdat. Otdel Akad. Nauk Moldav. SSR, Kishinev, 1970. MR281828