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...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....

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...⇒ 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...

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... 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...

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...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⇐⇒...

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..., (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...

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...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 ...

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... 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...

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...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 ...

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...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...