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A class of regular semigroups closed under taking direct products, regular subsemigroups, and homomorphic images is an existence-variety (or e-variety) of regular semigroups. Each e-variety of locally inverse semigroups can be characterized by a set of bi-identities. These are identities of terms of type $\langle 2, 2\rangle$ in two sorts of variables X and X'. In this paper we obtain a basis of bi-identities for the e-variety of locally inverse semigroups and for certain sub-e-varieties.

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... The binary relations on F(X) which are of the form = (V; X) for some class V of locally inverse semigroups are characterized as the bi-invariant congruences on F(X) which contain (LI; X). Here a bi-invariant congruence on F(X) is a congruence which satis es (1) x xx 0 x and x 0 x 0 xx 0 for all x 2 X; ...

... Let X be a xed countably in nite set and U ; V ; W be e-varieties of locally inverse semigroups. We are going to prove that (1) f (U~(V~W); X) g , f ((U~V)~W; X) g holds for arbitrary f; g 2 F(X). In order to do this, we rst have to analyze the expressions (U~(V~W); X) and ((U~V)~W; X). ...

... (gp ; x) = a (f) (p ; x) 7 ! (gp ; x) by (1) for and X instead of and X = a ~ (f) (p( ~ ); x) 7 ! ( (p) ; (p ; x)) (q ; x) 7 ! ...

We nd a multiplication on the lattice of existence varieties of locally inverse semigroups which extends the classical multiplication of group varieties and the recently found multiplication of inverse semigroup varieties. As a special case we get an associative multiplication on the lattice of varieties of completely simple semigroups.

... We shall recall the basic definitions and results necessary to understand the following treatment. For further information consult the papers [4,5,14,15,17,46]. ...

... As explained in [4,5], the adequate concept of equational theory for evarieties of locally inverse semigroups is based on the signature {·, ∧} and is with respect to a doubled alphabet X ∪ X ′ . Here X is, as usual, a countably infinite set of variables and X ′ = {x ′ | x ∈ X} is a disjoint copy of X; the elements of X ′ are devoted to represent inverses of the elements which are represented by the elements of X. ...

... The latter means that each bi-identity satisfied by all members of V can be derived, using natural deduction rules, from the bi-identities of B ∪ B(LI) where B(LI) is a basis for the bi-equational theory of the class of all locally inverse semigroups. A set consisting of four independent bi-identities which may serve as B(LI) has been found in [5]. For more analogues between the theory of e-varieties of regular semigroups and varieties of universal algebras see [4,5,17,46]. ...

We study matrix identities involving multiplication and unary operations such
as transposition or Moore-Penrose inversion. We prove that in many cases such
identities admit no finite basis.

... We collect some properties of the sandwich operation ∧ which partly can be found in [3]. Proposition 3. Let S be a locally inverse semigroup, and let s, t ∈ S. Then the following holds: ...

... Now we are ready to establish equality (3). For this let u, u 1 , u 2 , w ∈ s(F ), and let u be the inverse of u in the maximal subgroup G ab of BFCS(X). ...

In [Studia Sci. Math. Hungar. 41 (2004) 39-58] we constructed for a completely simple semigroup C an expansion , which is isomorphic to the Birget-Rhodes expansion [J. Algebra 120 (1989) 284-300], if C is a group. Analogous to the fact, proven in [J. Algebra 120 (1989) 284-300], that contains a copy of the free inverse semigroup in case C is the free group on X, we show that contains a copy of the bifree locally inverse semigroup, if C is the bifree completely simple semigroup on X. As a consequence, among other things, we obtain a new proof of a result due to F. Pastijn [Trans. Amer. Math. Soc. 273 (1982) 631-655] which says that each locally inverse semigroup divides a perfect rectangular band of E-unitary inverse monoids.

... This concept was the natural adaptation for e-varieties of the more common notion of free objects in varieties of algebras. Most paper at that time studied the structure of the bifree objects in e-varieties that have them, and tried to obtain Birkhoff type theorem for e-varieties [1,2,3,5,7,14,15,24] (see also [6,17,18,19]). ...

We prove the existence of a regular semigroup F(X) weakly generated by X such that all other regular semigroups weakly generated by X are homomorphic images of F(X). The semigroup F(X) is introduced by a presentation and the word problem for that presentation is solved.

... This theory began in the 1990s [15,17] and a great effort was made on the development of a Birkhoff-type theorem for e-varieties of regular semigroups. Unfortunately, only partial results were found, namely for the e-varieties of locally inverse semigroups [1,2] and for the e-varieties of regular E-solid semigroups [18], and the interest on general e-varieties of regular semigroups diminished considerably. These partial results were based on the concepts of 'bifree objects' and 'biequational classes'. ...

A regular semigroup is weakly generated by a set X if it has no proper regular subsemigroups containing X. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup FI(X) weakly generated by |X| idem-potents such that all other regular semigroups weakly generated by |X| idempotents are homomorphic images of FI(X). The semigroup FI(X) is defined by a presentation G(X), $\rho_e\cup\rho_s$ and its structure is studied. Although each of the sets G(X), $\rho_e$, and $\rho_s$ is infinite for |X| ≥ 2, we show that the word problem is decidable as each congruence class has a "canonical form". If $FI_n$ denotes FI(X) for |X| = n, we prove also that $FI_2$ contains copies of all $FI_n$ as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide $FI_2$; and (ii) all finite semigroups divide $FI_2$.

In this article we introduce the concept of a generalised existence variety of regular semigroups. This concept is analogous to that of a generalised variety of algebras of fixed finite type. We show that for regular semigroups that are E-solid or locally inverse, generalised existence varieties provide a link between e-varieties and e-pseudovarieties. Other results concerning e-varieties and e-pseudovarieties can then be obtained.

We employ the techniques developed in an earlier paper to show that involutory semigroups arising in various contexts do not have a finite basis for their identities. Among these are partition semigroups endowed with their natural inverse involution, including the full partition semigroup CnCn for n⩾2n⩾2, the Brauer semigroup BnBn for n⩾4n⩾4 and the annular semigroup AnAn for n⩾4n⩾4, n even or a prime power. Also, all of these semigroups, as well as the Jones semigroup JnJn for n⩾4n⩾4, turn out to be inherently nonfinitely based when equipped with another involution, the ‘skew’ one. Finally, we show that similar techniques apply to the finite basis problem for existence varieties of locally inverse semigroups.

A new model, in terms of finite bipartite graphs, of the free
pseudosemilattice is presented. This will then be used to obtain several
results about the variety SPS of all strict pseudosemilattices: (i) an identity
basis for SPS is found, (ii) SPS is shown to be inherently non-finitely based,
(iii) SPS is shown to have no irredundant identity basis, and (iv) SPS is shown
to have no covers and to be meet-prime in the lattice of all varieties of
pseudosemilattices. Some applications to e-varieties of locally inverse
semigroups are also derived.

It is proved that no non-orthodox e-variety ofE-solid semigroups has a finite basis for its bi-identities.

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