Interassociates of a free semigroup on two generators

A. B. Gorbatkov

For any semigroup \((S;\cdot)\) let \((S;\circ)\) be a semigroup defined on the same set. Semigroup \((S;\circ)\) is called an interassociate of \((S;\cdot)\) if the following identities hold \(x\cdot (y \circ z)=(x \cdot y) \circ z\) and \(x \circ (y \cdot z)=(x \circ y) \cdot z\). All interassociates of the free semigroup over the two-element alphabet are described.

1. B.N. Givens, K.A. Linton, A. Rosin, L. Dishman, Interassociates of the free commutative semigroup on n generators, Semigroup Forum, 74 (2007), 370–378.

2. A.B. Gorbatkov, Interassociativity on a free commutative semigroup, Siberian Math. J., 54 (2013), №3, 441–445.

3. M. Gould, K.A. Linton, A.W. Nelson, Interassociates of monogenic semigroups, Semigroup Forum, 68 (2004), 186–201.

4. J.B. Hickey, Semigroups under a sandwich operation, Proc. Edinburgh Math. Soc., 26 (1983), 371–382.

5. J.B. Hickey, On variants of a semigroup, Bull. Austral. Math. Soc., 34 (1986), 447–459.

6. T.A. Khan, M.V. Lawson, Variants of regular semigroups, Semigroup Forum, 62 (2001), 358–374.

7. G. Lallement, Semigroups and combinatorial applications, John Wiley & Sons, Inc., NY, 1979.

8. K.D. Magill, Semigroup structures for families of functions I. Some homomorphism theorems, J. Austral. Math. Soc., 7 (1967), 81–94.

9. A.V. Zhuchok, Commutative dimonoids, Algebra and Discrete Math., 2 (2009), 116–127.

10. A.V. Zhuchok, Free commutative dimonoids, Algebra and Discrete Math., 9 (2010), №1, 109–119.

11. D. Zupnik, On interassociativity and related questions, Aequationes Math., 6 (1971), 141–148.

	Interassociates of a free semigroup on two generators
Author	A. B. Gorbatkov gorbatkov_a@mail.ru Department of Mathematical Analysis and Algebra, Institute of Physics, Mathematics and Information Technologies, Luhansk Taras Shevchenko National University
Abstract	For any semigroup \((S;\cdot)\) let \((S;\circ)\) be a semigroup defined on the same set. Semigroup \((S;\circ)\) is called an interassociate of \((S;\cdot)\) if the following identities hold \(x\cdot (y \circ z)=(x \cdot y) \circ z\) and \(x \circ (y \cdot z)=(x \circ y) \cdot z\). All interassociates of the free semigroup over the two-element alphabet are described.
Keywords	free semigroup; interassociativity; interassociate of a semigroup
Reference	1. B.N. Givens, K.A. Linton, A. Rosin, L. Dishman, Interassociates of the free commutative semigroup on n generators, Semigroup Forum, 74 (2007), 370–378. 2. A.B. Gorbatkov, Interassociativity on a free commutative semigroup, Siberian Math. J., 54 (2013), №3, 441–445. 3. M. Gould, K.A. Linton, A.W. Nelson, Interassociates of monogenic semigroups, Semigroup Forum, 68 (2004), 186–201. 4. J.B. Hickey, Semigroups under a sandwich operation, Proc. Edinburgh Math. Soc., 26 (1983), 371–382. 5. J.B. Hickey, On variants of a semigroup, Bull. Austral. Math. Soc., 34 (1986), 447–459. 6. T.A. Khan, M.V. Lawson, Variants of regular semigroups, Semigroup Forum, 62 (2001), 358–374. 7. G. Lallement, Semigroups and combinatorial applications, John Wiley & Sons, Inc., NY, 1979. 8. K.D. Magill, Semigroup structures for families of functions I. Some homomorphism theorems, J. Austral. Math. Soc., 7 (1967), 81–94. 9. A.V. Zhuchok, Commutative dimonoids, Algebra and Discrete Math., 2 (2009), 116–127. 10. A.V. Zhuchok, Free commutative dimonoids, Algebra and Discrete Math., 9 (2010), №1, 109–119. 11. D. Zupnik, On interassociativity and related questions, Aequationes Math., 6 (1971), 141–148.
Pages	139-145
Volume	41
Issue	2
Year	2014
Journal	Matematychni Studii
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