## PARTIAL ORDERS IN REGULAR SEMIGROUPS (2010)

### Citations

566 |
Fundamentals of semigroup theory
- Howie
- 1995
(Show Context)
Citation Context ...ehasa = bsinceNisanti − symmetric.For(a, b), (b, c) ∈ Sanda = axa, ax = xaforsomex ∈ Swehavea = a2x = baxandthusca = cbax = b2ax = ba2x = ba = a2andsimilarlyac = a2, whichshows(a, c) ∈ S. Recall from =-=[8]-=-, p. 222, that a regular semigroup (S, .) in which the set of idempotents E(S) is a subsemigroup is called orthodox, and if E(S) is a normal band then (S, .) is called a generalized inverse semigroup.... |

153 | Lectures on Semigroups - Petrich - 1977 |

35 |
Congruences on completely regular semigroups, preprint
- Petrich
(Show Context)
Citation Context ...⇒ a = axa = axb = bxa for some x ∈ S, (a, b) ∈ N1 ⇐⇒ a = xa = xb = by for some x, y ∈ S, (a, b) ∈ S ⇐⇒ a2 = ab = ba. Obviously, N ⊆ N1, and easy calculations show that N1 is a partial order on S (cf. =-=[6]-=-). The relation C was defined in [1] by Conrad, and it was proved in [2] by Burgess and Raphael (cf. Lemma 1) that C is a partial order on S iff (S, .) is weakly separative, i. e., for all a, b ∈ S as... |

15 |
The natural partial order on a regular semigroup
- Nambooripad
- 1980
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Citation Context ... and it was proved in [2] by Burgess and Raphael (cf. Lemma 1) that C is a partial order on S iff (S, .) is weakly separative, i. e., for all a, b ∈ S asa = asb = bsa = bsbforalls ∈ Simpliesa = b. In =-=[3]-=-, Nambooripad defined N and showed that it is a partial order iff (S, .) is regular (cf. Lemma 2). The relation S was introduced by Drazin in [4], and he proved that, for any completely regular semigr... |

2 |
A partial order in completely regular semigroups
- Drazin
- 1986
(Show Context)
Citation Context ...b ∈ S asa = asb = bsa = bsbforalls ∈ Simpliesa = b. In [3], Nambooripad defined N and showed that it is a partial order iff (S, .) is regular (cf. Lemma 2). The relation S was introduced by Drazin in =-=[4]-=-, and he proved that, for any completely regular semigroup, S is a partial order on S, and C ⊆ S ⊆ N holds (cf. Lemma 3). Finally, for any regular semigroup (S, .), the natural partial order ≤ a ≤ b⇐⇒... |

1 |
The hulls of semiprime rings
- Conrad
(Show Context)
Citation Context ..., (a, b) ∈ N1 ⇐⇒ a = xa = xb = by for some x, y ∈ S, (a, b) ∈ S ⇐⇒ a2 = ab = ba. Obviously, N ⊆ N1, and easy calculations show that N1 is a partial order on S (cf. [6]). The relation C was defined in =-=[1]-=- by Conrad, and it was proved in [2] by Burgess and Raphael (cf. Lemma 1) that C is a partial order on S iff (S, .) is weakly separative, i. e., for all a, b ∈ S asa = asb = bsa = bsbforalls ∈ Simplie... |

1 |
On Conrads partial order relation on semiprime rings and semigroups, Semigroup Forum 16
- Burgess, Raphael
- 1978
(Show Context)
Citation Context ...r some x, y ∈ S, (a, b) ∈ S ⇐⇒ a2 = ab = ba. Obviously, N ⊆ N1, and easy calculations show that N1 is a partial order on S (cf. [6]). The relation C was defined in [1] by Conrad, and it was proved in =-=[2]-=- by Burgess and Raphael (cf. Lemma 1) that C is a partial order on S iff (S, .) is weakly separative, i. e., for all a, b ∈ S asa = asb = bsa = bsbforalls ∈ Simpliesa = b. In [3], Nambooripad defined ... |

1 |
Some partial orders on completely regular semigroups
- Liu, Song
- 2004
(Show Context)
Citation Context ... a partial order on S, and C ⊆ S ⊆ N holds (cf. Lemma 3). Finally, for any regular semigroup (S, .), the natural partial order ≤ a ≤ b⇐⇒ a = be = fbforsomee, f ∈ E(S) satisfies N ⊆≤ . It was shown in =-=[5]-=- that for any completely regular semigroup (S, .) the relations S and ≤ coincide iff S is a cryptogroup, i. e., Greens H-relation is a congruence on (S, .), and that C and S coincide iff S is a normal... |

1 |
An Introduction to the Structure Theory, Pure and Applied Mathematics 193
- Grillet, Semigroups
- 1995
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Citation Context ...the class 196 K. V. R. Srinivas and Y. L. Anasuya of all orthodox semigroups and the class of all locally inverse semigroups. For the latter we have the following characterization (due to Lemma 2 and =-=[9]-=-, IX. Proposition 3.2): Theorem 4. Let (S, .) be a regular semigroup. Then the following statements are equivalent: a) (S, .) is locally inverse; b) N is compatible; c) If aNb and b0 is an inverse of ... |

1 |
Characterization of partial orders on regular semigroups, A. P. Akademi of sciences
- Murthy, V, et al.
- 2004
(Show Context)
Citation Context ...igroup. Then the following statements are equivalent: a) (S, .) is locally inverse; b) N is compatible; c) If aNb and b0 is an inverse of b, then there is a unique inverse a0 of a such that a0Nb0. In =-=[10]-=-, the following lemma was proved. Lemma 5 : Let (S, .) be a locally inverse semigroup and ≤ a partial order on S with the following properties: (i) ≤ is compatible with the multiplication; (ii) the re... |