Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 2, 569-588.

Relatively Free *-Bands

Mario Petrich^{1} Pedro V. Silva^{2}

Centro de Matemática, Faculdade de Ciências, Universidade do Porto, 4050 Porto, Portugal http://www.fc.up.pt/cmup

Abstract. A *-band is an algebra consisting of a band (idempotent semigroup) on which an involution * is defined satisfying an extra condition; in summary

(xy)^{∗} = y^{∗}x^{∗}, x^{∗∗} = x, x = xx^{∗}x, (xy)z = x(yz), x^{2} = x.

The lattice of all *-band varieties was determined by Adair who also provided a basis for the identities of each variety. Another system of bases was devised by Petrich. Defining certain operators on the free involutorial semigroup F on a nonempty set X, we construct a system of fully invariant congruences on F which is in bijection with the set of all proper *-band varieties, with the exception of normal *-band varieties which require a different treatment. The proof of this result is based on those evoked above and is broken into a long sequence of lemmas. MSC 2000: 20M05, 20M07

1. Introduction and summary

Varieties of bands (idempotent semigroups) have attracted considerable attention and with good reason. Their lattice has been determined representing a prime example of mathematical ingenuity and success. The varieties themselves admit copious characterizations and their members convenient structure theorems. For example, free objects in these varieties have been described in several ways exhibiting a relatively transparent structure.

1Supported by F.C.T. and PRAXIS XXI/BCC/4358/94 2Supported by F.C.T. and Project AGC/PRAXIS XXI/2/2.1/MAT/63/94 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

As a variant of a general semigroup and inspired by ring theory, emerged a *-semigroup, which is an algebra consisting of a semigroup on which an involution is defined. As a natural special case, we have a *-band as a band and a *-semigroup with the involution satisfying an additional “regularity” condition. All together, we have a mapping x → x^{∗} of a semigroup S into itself satisfying the following axioms:

(xy)^{∗} = y^{∗}x^{∗}, x^{∗∗} = x, x = xx^{∗}x, x^{2} = x.  (1)

The lattice of *-band varieties was determined by Adair [1] based on the approach of Fennemore [2] in describing the lattice of band varieties. Adair also provided a system of bases for identities for each *-band variety. As proved in [5], the system of identities devised in [4] for join irreducible band varieties can be used also for *-band varieties. As a consequence, we have that *-band varieties admit bases for their identities consisting of starfree identities. Even though *-band varieties (as well as varieties of band monoids) are treated as an afterthought of band varieties, the deliberations involved turn out quite nontrivial and intrinsically interesting.

Yamada [8] characterized finitely generated free *-bands using an inductive process. The purpose of this paper is to construct free objects in every *-band variety on any nonempty set X by solving the corresponding word problem on the free involutorial semigroup. Section 2 contains a minimum of needed notation and terminology. For varieties different from B, the variety of all *-bands, and varieties of normal *-bands, our treatment consists of three stages. In the first stage, Section 3, we introduce operators h_{n} and i_{n} on the free involutorial semigroup F on X and establish some of their properties. In the second stage, Section 4, we use these operators to introduce operators γ_{t_{n}} with t ∈ {h, i} and n ≥ 3 and prove a number of their properties which culminate in the fact that they induce fully invariant *-band congruences on F. The third stage, Section 5, consists in identifying the *-band varieties which correspond to these congruences. In Section 6 the main theorem provides free objects for all nontrivial *-band varieties.

2. Notation and terminology

For these, we follow the standard texts in semigroups and [4] with the following supplements.

If Y is a set, | Y | stands for the cardinality of Y. We fix a nonempty set X and consider a bijection x → x^{∗} of X onto a disjoint copy X^{∗} of X. Let I = X ∪ X^{∗} and F be the free semigroup on I, which consists of all nonempty words over the alphabet I. We may view F as an involutory semigroup by defining

(x^{∗})^{∗} = x, (y_{1} … y_{n})^{∗} = y_{n}^{∗} … y_{1}^{∗}

for all x ∈ X, n ≥ 2 and y_{1}, …, y_{n} ∈ I. We denote by F^{1} the free monoid on I, obtained by adjoining the empty word 1 to F.

For emphasis, we refer to a homomorphism of *-semigroups as a *-homomorphism and the induced congruence as a *-congruence. In this terminology, the terms homomorphism and congruence refer to multiplication alone. We denote

End(F) – the set of all *-endomorphisms of F.

For an identity u = v on *-bands, we denote by [u = v] the variety of *-bands determined by u = v. We omit the covering identities (1).

In the treatment of relatively free bands in [6], we have used a number of invariants and operators on the set of words over a nonempty set X, c(w) for content of w, ♯(w) for cardinality of c(w), s(w), σ(w), ε(w), e(w). In the context of *-bands when we are concerned with I = X ∪ X^{∗} rather than with X alone, we must make suitable adjustments in the definitions of these functions relative to the occurrence of some starred letters. It is natural then to denote the new function by p^{∗} if it represents a modification of the previously used symbol p relative to X. For typographical, as well as esthetical, reasons we shall use the same notation p as for X but in the context of *-bands. This should cause no confusion as to the meaning of the symbol p since it will be precisely defined and the original ones relative to X will not be used.

Now let w ∈ F; then

c(w) – the set of all letters x ∈ X such that either x or x^{∗} occurs in w, c(1) = ∅ (in [3], the notation c_{X}(w) is used),

♯(w) = | c(w) |,

u – a prefix of w if w = uv for some v ∈ F^{1},

\overline{w} – the word obtained from w by reversing the order of letters, that is if w = x_{1}x_{2} … x_{n}, then \overline{w} = x_{n} … x_{2}x_{1}, \overline{1} = 1,

s(w) and σ(w) – there is a unique factorization w = uyv with u, v ∈ F^{1}, y ∈ I and c(u) ⊂ c(uy) = c(w); we write s(w) = u and σ(w) = y (in [3], the notation s_{X}(w) and σ_{X}(w) is used),

e(w) – the left-right dual of s(w), ε(w) – the left-right dual of σ(w). For any operator t on F, set t(1) = 1 thereby extending it to F^{1}, and define operators \overline{t}

and t^{∗} on F by

\overline{t}(w) = \overline{t(\overline{w})}, t^{∗}(w) = (t(w^{∗}))^{∗}.

For any w ∈ F, we have \overline{\overline{w}} = w^{∗∗} = w and thus \overline{\overline{t}} = t^{∗∗} = t. We shall need the following simple result.

Lemma 2.1. We have \overline{s} = s^{∗} = e, \overline{σ} = σ^{∗} = ε and σ(w) = ε(\overline{w}), ε(w) = σ(\overline{w}) for all x ∈ F.

Proof. Let w ∈ F. Then there exist u, v ∈ F^{1} and y ∈ I such that w = uyv and c(v) ⊂ c(yv) = c(w). Hence

\begin{aligned}\overline{s}(w) & = & \overline{s(\overline{w})} = \overline{s(\overline{v} \overline{y} \overline{u})} = \overline{\overline{v}} = v = e(w), \\ \overline{σ}(w) & = & \overline{σ(\overline{w})} = \overline{σ(\overline{v} \overline{y} \overline{u})} = \overline{\overline{y}} = y = ε(w) \end{aligned}

and the proof for star is the same.

The last assertions of the lemma are obvious. □

Besides the results in Lemma 2.1, we shall see in the next section that \overline{t} = t^{∗} for the operators t which are of central importance in our deliberations. We thus could either use \overline{t} or t^{∗} throughout; our choice is \overline{t} for typographical reasons.

3. Operators h_{n} and i_{n}

This section consists of lemmas which will be used in the next section. We start by introducing the needed notation. Let w ∈ F.

If w = yz with y ∈ I and z ∈ F^{1}, we write h_{2}(w) = y. The operator i_{2} is defined on F inductively on ♯(w) by the formula

i_{2}(w) = i_{2}s(w) σ(w).

Hence i_{2}(w) is the word obtained from w retaining only the first occurrence of each letter regarding x and x^{∗} as the same letter.

The next set of operators is also defined on F inductively: for t ∈ {h, i} and n > 2, let

t_{n}(w) = t_{n}s(w) σ(w) \overline{t_{n−1}}(w).  (2)

It is important to note that this formula harbors two inductions: one is on n and the other is on ♯(w). For ♯s(w) = ♯(w) − 1 unless w = 1. The proofs will generally be by (the main) induction on n and occasionally, for the first and/or the inductive step, also by induction on ♯(w).

In several proofs by induction, the following notation will come in handy:

i_{1}(w) = 1 (w ∈ F),  χ_{n} = h_{n+1} (n ≥ 1).  (3)

This device will make it possible to start the induction process at n = 1. The case χ_{1} = h_{2} is generally easy to check while the instance i_{1} usually holds trivially. Observe that the inductive formula (2) remains valid for operators χ_{n} and i_{n} for n > 1.

In this section t ∈ {χ, i}.

Lemma 3.1. For n ≥ 2 and w ∈ F, we have

\overline{t_{n}}(w) = t_{n−1}(w) ε(w) \overline{t_{n}}e(w).

Proof. Direct application of definitions yields

\begin{aligned}\overline{t_{n}}(w) & = & \overline{t_{n}(\overline{w})} = \overline{t_{n}s(\overline{w}) σ(\overline{w}) \overline{t_{n−1}}(\overline{w})} = \overline{\overline{t_{n−1}}(\overline{w})} σ(\overline{w}) \overline{t_{n}s(\overline{w})} \\ & = & t_{n−1}(w) \overline{σ}(w) \overline{t_{n}}\overline{s(\overline{w})} = t_{n−1}(w) ε(w) \overline{t_{n}}e(w). \end{aligned}

□

Lemma 3.2. For n ≥ 2, we have ct_{n} = c, st_{n} = t_{n}s, σt_{n} = σ.

Proof. We use induction on n. Let n ≥ 2 and assume that the lemma holds for t_{m} whenever 2 ≤ m < n. We use a secondary induction on ♯(w), where w ∈ F^{1}.

All equalities hold trivially for w = 1. Let w ∈ F and assume the equalities hold for t_{n} and z ∈ F^{1} whenever ♯(z) < ♯(w). Recall formula (2). By induction hypothesis on ♯(w), we have

ct_{n}s(w) = cs(w)  (4)

and thus

c(t_{n}s(w) σ(w)) = c(s(w) σ(w)) = c(w).  (5)

If n = 2, then c\overline{t_{n−1}}(w) ⊆ c(w) follows from the definition of t_{1}. Otherwise, we get

c\overline{t_{n−1}}(w) = c(\overline{t_{n−1}(\overline{w})}) = ct_{n−1}(\overline{w}) = c(\overline{w}) = c(w)

by induction on n. Either way, it follows that ct_{n}(w) = c(w) and so ct_{n} = c. From (4) and (5) we deduce that st_{n}(w) = t_{n}s(w) and σt_{n}(w) = σ(w). □

Lemma 3.3. For n ≥ 2 and w = uyv where u, v ∈ F^{1}, y ∈ I and c(y) ∩ c(u) = ∅, we have t_{n}(w) ∈ t_{n}(u) yF^{1}.

Proof. The argument is by induction on ♯(w). Assume that the lemma holds for all q ∈ F such that 1 ≤ ♯(q) < ♯(w). If u = s(w), then y = σ(w) and so

t_{n}(w) = t_{n}s(w) σ(w) \overline{t_{n−1}}(w) ∈ t_{n}(u) yF^{1}.

Assume that u ≠ s(w). Since ♯(u) < ♯(w), we may write s(w) = uyr for some r ∈ F^{1}. By our induction hypothesis, we have t_{n}s(w) = t_{n}(u) yz for some z ∈ F^{1} and thus

t_{n}(w) = t_{n}s(w) σ(w) \overline{t_{n−1}}(w) = t_{n}(u) yz σ(w) \overline{t_{n−1}}(w) ∈ t_{n}(u) yF^{1}.  □

Lemma 3.4. For n ≥ 1, we have \overline{t_{n}} = t_{n}^{∗}.

Proof. We use induction on n. Let w ∈ F.

Let n = 1. The case t = i being trivial, we assume that t = χ. If w = zy with y ∈ I and z ∈ F^{1}, then

\overline{χ_{1}}(w) = \overline{χ_{1}(\overline{w})} = χ_{1}(y\overline{z}) = y = (y^{∗})^{∗} = (χ_{1}(y^{∗}z^{∗}))^{∗} = (χ_{1}(w^{∗}))^{∗} = χ_{1}^{∗}(w).

Let n > 1 and assume that the result holds for n − 1.

We use secondary induction on ♯(w). Assume that ♯(w) > 0 and \overline{t_{n}}(z) = t_{n}^{∗}(z) whenever ♯(z) < ♯(w). Hence

\overline{t_{n}}(w) = t_{n−1}(w) ε(w) \overline{t_{n}}e(w)  by Lemma 3.1
= t_{n−1}(w) ε(w) t_{n}^{∗}e(w)  by induction on ♯(w)
= t_{n−1}(w) σ^{∗}(w) t_{n}^{∗}s^{∗}(w)  by Lemma 2.1
= t_{n−1}(w) σ^{∗}(w) t_{n}^{∗}((s(w^{∗}))^{∗}) = (t_{n−1}^{∗}(w^{∗}))^{∗} (σ(w^{∗}))^{∗} (t_{n}s(w^{∗}))^{∗}
= (t_{n}s(w^{∗}) σ(w^{∗}) t_{n−1}^{∗}(w^{∗}))^{∗}
= (t_{n}s(w^{∗}) σ(w^{∗}) \overline{t_{n−1}}(w^{∗}))^{∗}  by induction on n
= (t_{n}(w^{∗}))^{∗} = t_{n}^{∗}(w). □

Lemma 3.5. For n ≥ 1, we have χ_{1} = χ_{1}χ_{n} = χ_{1}\overline{χ_{n+1}}.

Proof. Since

\overline{χ_{n+1}}(w) = χ_{n}(w) ε(w) \overline{χ_{n+1}}e(w)

by Lemma 3.1, it is enough to show that χ_{1}χ_{n}(w) = χ_{1}(w) for all n ≥ 1 and w ∈ F. This follows from Lemma 3.3 by taking u = 1. □

Lemma 3.6. For m ≥ n ≥ 1 and u, v, w ∈ F^{1}, we have

t_{n}(u t_{m}(v) w) = t_{n}(u \overline{t_{m+1}}(v) w) = t_{n}(uvw).

Proof. We use induction on n. Let n = 1. Since the case t = i is trivial, we may assume that t = χ. Without loss of generality, we may assume that u = 1 and v ≠ 1. It follows from Lemma 3.5 that

χ_{1}(u χ_{m}(v) w) = χ_{1}χ_{m}(v) = χ_{1}(v) = χ_{1}(uvw)

and the equality χ_{1}(u \overline{χ_{m+1}}(v) w) = χ_{1}(uvw) is obtained similarly.

Now let n > 1 and assume that the lemma holds for n − 1. First we show that

\overline{t_{n−1}}(u t_{m}(v) w) = \overline{t_{n−1}}(u \overline{t_{m+1}}(v) w) = \overline{t_{n−1}}(uvw)  (6)

for all m ≥ n and u, v, w ∈ F^{1}. Indeed,

\overline{t_{n−1}}(u t_{m}(v) w) = \overline{t_{n−1}(\overline{u t_{m}(v) w})} = \overline{t_{n−1}(\overline{w} \overline{t_{m}(v)} \overline{u})}
= \overline{t_{n−1}(\overline{w} \overline{t_{m}}(\overline{v}) \overline{u})} = \overline{t_{n−1}(\overline{w} \overline{v} \overline{u})}  by induction on n
= \overline{t_{n−1}(\overline{uvw})} = \overline{t_{n−1}}(uvw),

and the equality t_{n−1}(u \overline{t_{m+1}}(v) w) = t_{n−1}(uvw) is proved similarly.

Next we show that

t_{n}(u t_{m}(v)) = t_{n}(u \overline{t_{m+1}}(v)) = t_{n}(uv)  (7)

for all m ≥ n and u, v ∈ F^{1} by a secondary induction on d = | c(v) \ c(u) |.

Let d = 0, that is, c(v) ⊆ c(u). Then

t_{n}(u t_{m}(v)) = t_{n}s(u t_{m}(v)) σ(u t_{m}(v)) \overline{t_{n−1}}(u t_{m}(v))
= t_{n}s(u) σ(u) \overline{t_{n−1}}(u t_{m}(v))  since d = 0
= t_{n}s(u) σ(u) \overline{t_{n−1}}(uv)  by (6)
= t_{n}s(uv) σ(uv) \overline{t_{n−1}}(uv)  since d = 0
= t_{n}(uv)

and the equality t_{n}(u \overline{t_{m+1}}(v)) = t_{n}(uv) is proved similarly.

Now let d > 0 and assume that (7) holds for all values smaller than d. We can write v = zyr with uz = s(uv) and y = σ(uv). By Lemma 3.3, we have t_{m}(v) ∈ t_{m}(z) yF^{1} and so Lemma 3.2 yields

s(u t_{m}(v)) = u t_{m}(z),  σ(u t_{m}(v)) = y.  (8)

Thus

t_{n}(u t_{m}(v)) = t_{n}s(u t_{m}(v)) σ(u t_{m}(v)) \overline{t_{n−1}}(u t_{m}(v))
= t_{n}(u t_{m}(z)) y \overline{t_{n−1}}(u t_{m}(v))  by (8)
= t_{n}(uz) y \overline{t_{n−1}}(u t_{m}(v))  by induction on d
= t_{n}s(uv) σ(uv) \overline{t_{n−1}}(uv)  by (6)
= t_{n}(uv).

Since by Lemma 3.1 we have

\overline{t_{m+1}}(v) = t_{m}(v) ε(v) \overline{t_{m+1}}e(v),

Lemma 3.2 yields that

s(u \overline{t_{m+1}}(v)) = s(u t_{m}(v)), σ(u \overline{t_{m+1}}(v)) = σ(u t_{m}(v))

and so the equality t_{n}(u \overline{t_{m+1}}(v)) = t_{n}(uv) follows easily from the above. Thus (7) holds.

We now show that

t_{n}(u t_{m}(v) w) = t_{n}(uvw)  (9)

for all m ≥ n and u, v, w ∈ F^{1} by induction on e = | c(w) \ c(uv) |. Let m ≥ n, u, v, w ∈ F^{1} and r = u t_{m}(v) w.

Let e = 0, that is, c(w) ⊆ c(uv). Then

t_{n}(r) = t_{n}s(r) σ(r) \overline{t_{n−1}}(r)
= t_{n}s(u t_{m}(v)) σ(u t_{m}(v)) \overline{t_{n−1}}(r)  since e = 0
= st_{n}(u t_{m}(v)) σt_{n}(u t_{m}(v)) \overline{t_{n−1}}(r)  by Lemma 3.2
= st_{n}(uv) σt_{n}(uv) \overline{t_{n−1}}(r)  by (7)
= t_{n}s(uv) σ(uv) \overline{t_{n−1}}(r)  by Lemma 3.2
= t_{n}s(uvw) σ(uvw) \overline{t_{n−1}}(r)  since e = 0
= t_{n}s(uvw) σ(uvw) \overline{t_{n−1}}(uvw)  by (6)
= t_{n}(uvw).

The equality t_{n}(u \overline{t_{m+1}}(v) w) = t_{n}(uvw) is proved similarly and so the lemma holds when e = 0.

Let e > 0 and assume that (9) holds for all values smaller than e. Let m ≥ n. Since c(w) ⊈ c(uv), we may write w = zyq with uvz = s(uvw) and y = σ(uvw). In particular, we have | c(z) \ c(uv) | < | c(w) \ c(uv) |. By Lemma 3.2, we have ct_{m}(v) = c(v) and hence

t_{n}(r) = t_{n}s(r) σ(r) \overline{t_{n−1}}(r) = t_{n}(u t_{m}(v) z) y \overline{t_{n−1}}(r)
= t_{n}(uvz) y \overline{t_{n−1}}(r)  by induction on e
= t_{n}s(uvw) σ(uvw) \overline{t_{n−1}}(uvw)  by (6)
= t_{n}(uvw).

This proves relation (9). The equality t_{n}(u \overline{t_{m+1}}(v) w) = t_{n}(uvw) is proved similarly. □

Lemma 3.7. Let φ ∈ End(F). Then \overline{φ} ∈ End(F).

Proof. For all u, v ∈ F, we have

\overline{φ}(uv) = \overline{φ(\overline{uv})} = \overline{φ(\overline{v} \overline{u})} = \overline{φ(\overline{v}) φ(\overline{u})} = \overline{φ(\overline{u})} \overline{φ(\overline{v})} = \overline{φ}(u) \overline{φ}(v).

Since \overline{w}^{∗} = \overline{w^{∗}} for every w ∈ F, we also obtain

\overline{φ}(w^{∗}) = \overline{φ(\overline{w^{∗}})} = \overline{φ(\overline{w}^{∗})} = \overline{(φ(\overline{w}))^{∗}} = \overline{φ(\overline{w})}^{∗} = (\overline{φ}(w))^{∗}

and so \overline{φ} ∈ End(F). □

Lemma 3.8. Let n ≥ 1, u, v, w, z ∈ F^{1} and φ ∈ End(F) be such that t_{n}(u) = t_{n}(v). Then t_{n}(w φ(u) z) = t_{n}(w φ(v) z).

Proof. We use induction on n. Let n = 1. Since the case t = i is trivial, we may assume that t = χ. Without loss of generality, we may assume that w = 1 and u, v ≠ 1. By hypothesis, we have χ_{1}(u) = χ_{1}(v) = y for some y ∈ I. Thus

χ_{1}(w φ(u) z) = χ_{1}φ(u) = χ_{1}φ(y) = χ_{1}φ(v) = χ_{1}(w φ(v) z)

and the lemma holds for n = 1.

Assume now that n > 1 and that the lemma holds for n − 1. Note that t_{n}(u) = t_{n}(v) implies

\begin{aligned}t_{n−1}(\overline{u}) & = & t_{n−1}\overline{t_{n}}(\overline{u}) by Lemma 3.6 \\ & = & t_{n−1}\overline{t_{n}(u)} = t_{n−1}\overline{t_{n}(v)} = t_{n−1}\overline{t_{n}}(\overline{v}) = t_{n−1}(\overline{v}). \end{aligned}

For all u, v, w, z ∈ F^{1}, we get

\overline{t_{n−1}}(w φ(u) z) = \overline{t_{n−1}(\overline{w φ(u) z})} = \overline{t_{n−1}(\overline{z} \overline{φ(u)} \overline{w})}
= \overline{t_{n−1}(\overline{z} \overline{φ}(\overline{u}) \overline{w})} = \overline{t_{n−1}(\overline{z} \overline{φ}(\overline{v}) \overline{w})}

by the above, Lemma 3.7 and induction on n

= \overline{t_{n−1}(\overline{z} \overline{φ(v)} \overline{w})} = \overline{t_{n−1}(\overline{w φ(v) z})} = \overline{t_{n−1}}(w φ(v) z).

It follows that

\overline{t_{n−1}}(w φ(u) z) = \overline{t_{n−1}}(w φ(v) z).  (10)

We use secondary induction on ♯(u). The case ♯(u) = 0 being trivial, let ♯(u) > 0 and assume that t_{n}(w φ(u′) z) = t_{n}(w φ(v′) z) whenever t_{n}(u′) = t_{n}(v′) and ♯(u′) < ♯(u).

Now we introduce tertiary induction on d = | c(z) \ c(w φ(u)) |.

Let d = 0, that is, c(z) ⊆ c(w φ(u)). Write r = w φ(u) z. If cφ(u) ⊆ c(w), then

t_{n}(r) = t_{n}s(r) σ(r) \overline{t_{n−1}}(r)
= t_{n}s(w) σ(w) \overline{t_{n−1}}(r)  since cφ(u) ⊆ c(w)
= t_{n}s(w) σ(w) \overline{t_{n−1}}(w φ(v) z)  by (10)
= t_{n}(w φ(v) z)  by symmetry.

Therefore we assume that cφ(u) ⊈ c(w). Then we can write u = u′yu′′ and φ(y) = qxq′ so that

s(w φ(u) z) = w φ(u′) q,  σ(w φ(u) z) = x.  (11)

Similarly, we have a factorization v = v′y′v′′ such that

c(w φ(v′)) ⊂ c(w φ(v′y′)) = c(w φ(u) z).

It follows easily from Lemma 3.3 that i_{2}t_{n}(a) = i_{2}(a) for every a ∈ F^{1}. Since t_{n}(u) = t_{n}(v), we get

i_{2}(u) = i_{2}t_{n}(u) = i_{2}t_{n}(v) = i_{2}(v).

This yields that y′ = y. In particular, we have

s(w φ(v) z) = w φ(v′) q, σ(w φ(v) z) = x.

By Lemma 3.3, we obtain

t_{n}(u) ∈ t_{n}(u′) yF^{1}, t_{n}(v) ∈ t_{n}(v′) yF^{1}.

Since t_{n}(u) = t_{n}(v) and y ∉ c(u′) ∪ c(v′), we must have

t_{n}(u′) = t_{n}(v′).  (12)

Therefore

t_{n}(r) = t_{n}s(r) σ(r) \overline{t_{n−1}}(r)
= t_{n}(w φ(u′) q) x \overline{t_{n−1}}(r)  by (11)
= t_{n}(w φ(v′) q) x \overline{t_{n−1}}(r)  by (12) and induction on ♯(u)
= t_{n}(w φ(v′) q) x \overline{t_{n−1}}(w φ(v) z)  by (10)
= t_{n}(w φ(v) z)  by symmetry,

proving the case d = 0.

Finally, assume that d > 0 and t_{n}(w φ(u) z′) = t_{n}(w φ(v) z′) whenever |c(z′)\c(w φ(u))|< d. Then we can write z = qyq′ with w φ(u) q = s(w φ(u) z) and y = σ(w φ(u) z). Since t_{n}(u) = t_{n}(v), Lemma 3.2 yields that cφ(u) = cφ(v) and so also w φ(v) q = s(w φ(v) z) and y = σ(w φ(v) z). Thus

t_{n}(r) = t_{n}s(r) σ(r) \overline{t_{n−1}}(r) = t_{n}(w φ(u) q) y \overline{t_{n−1}}(r)
= t_{n}(w φ(v) q) y \overline{t_{n−1}}(r)  by induction on d
= t_{n}s(w φ(v) z) σ(w φ(v) z) \overline{t_{n−1}}(w φ(v) z)  by (10)
= t_{n}(w φ(v) z)

completing the inductive step for all three inductions. □

4. Operators γ_{t_{n}}

The purpose of this section is to construct a family of fully invariant *-band congruences on F. We start by introducing the needed notation. For t ∈ {χ, i} and n ≥ 3, we define an operator γ_{t_{n}} on F by

γ_{t_{n}}(w) = t_{n−1}s(w) σ(w) ε(w) \overline{t_{n−1}}e(w).

Also let

Γ = {γ_{t_{n}} | t ∈ {χ, i}, n ≥ 3}.

Note that the operator γ_{t_{n}} is similar to the operator θ_{pq} introduced in ([7], Section 3) with the important difference that γ_{t_{n}} is defined by means of t_{n−1} and not by t_{n}.

Lemma 4.1. For γ ∈ Γ, we have γ = \overline{γ} = γ^{∗} = γ^{2}.

Proof. Let γ = γ_{t_{n}} and w ∈ F. Then

\overline{γ}(w) = \overline{γ(\overline{w})} = \overline{t_{n−1}s(\overline{w}) σ(\overline{w}) ε(\overline{w}) \overline{t_{n−1}}e(\overline{w})}
= \overline{\overline{t_{n−1}}e(\overline{w})} \overline{ε(\overline{w})} \overline{σ(\overline{w})} \overline{t_{n−1}s(\overline{w})}
= t_{n−1}(\overline{e(\overline{w})}) \overline{ε}(w) \overline{σ}(w) \overline{t_{n−1}}(\overline{s(\overline{w})})
= t_{n−1}\overline{e}(w) \overline{ε}(w) \overline{σ}(w) \overline{t_{n−1}} \overline{s}(w)
= t_{n−1}s(w) σ(w) ε(w) \overline{t_{n−1}}e(w)  by Lemma 2.1
= γ(w)

so that \overline{γ} = γ; the argument for the star is the same. Next

γ^{2}(w) = t_{n−1}sγ(w) σγ(w) εγ(w) \overline{t_{n−1}}eγ(w)
= t_{n−1}t_{n−1}s(w) σ(w) ε(w) \overline{t_{n−1}}\overline{t_{n−1}}e(w)
= t_{n−1}s(w) σ(w) ε(w) \overline{t_{n−1}}e(w)  by Lemma 3.6
= γ(w)

and thus also γ^{2} = γ. □

Lemma 4.2. For t ∈ {χ, i}, 1 ≤ n < m ≥ 3 and u, v, w ∈ F^{1}, we have

t_{n}(u γ_{t_{m}}(v) w) = t_{n}(uvw).

Proof. We use induction on n.

Let n = 1. Without loss of generality, we may assume that t = χ, u = 1 and v ≠ 1. It follows that

χ_{1}(u γ_{t_{m}}(v) w) = χ_{1}(h_{m−1}s(v) σ(v))
= χ_{1}(s(v) σ(v))  by Lemma 3.6
= χ_{1}(uvw).

Now let n > 1 and assume that the lemma holds for all values smaller than n. We use secondary induction on ♯(uvw). The case ♯(uvw) = 0 being trivial, let ♯(uvw) = k > 0 and assume that t_{n}(u′ γ_{t_{m}}(v′) w′) = t_{n}(u′v′w′) whenever ♯(u′v′w′) < k. We have

\overline{t_{n−1}}(u γ_{t_{m}}(v) w) = \overline{t_{n−1}(\overline{u γ_{t_{m}}(v) w})} = \overline{t_{n−1}(\overline{w} \overline{γ_{t_{m}}(v)} \overline{u})}
= \overline{t_{n−1}(\overline{w} γ_{t_{m}}(\overline{v}) \overline{u})}  by Lemma 4.1
= \overline{t_{n−1}(\overline{w} \overline{v} \overline{u})}  by induction on n
= \overline{t_{n−1}(\overline{uvw})} = \overline{t_{n−1}}(uvw)

and thus

\overline{t_{n−1}}(u γ_{t_{m}}(v) w) = \overline{t_{n−1}}(uvw).  (13)

Now we consider two cases.

\underline{Case}: ♯(uv) < k. Let uvz = s(uvw), y = σ(uvw) and r = u γ_{t_{m}}(v) w. Then

t_{n}(r) = t_{n}s(r) σ(r) \overline{t_{n−1}}(r)
= t_{n}(u γ_{t_{m}}(v) z) y \overline{t_{n−1}}(r)  since cγ_{t_{m}}(v) = c(v)
= t_{n}(uvz) y \overline{t_{n−1}}(r)  by induction on ♯(uvw)
= t_{n}s(uvw) σ(uvw) \overline{t_{n−1}}(uvw)  by (13)
= t_{n}(uvw).

\underline{Case}: ♯(uv) = k. By Lemma 3.6, we have

t_{n}(u t_{m−1}s(v) σ(v)) = t_{n}(u s(v) σ(v)).

Write p = u t_{m−1}s(v) σ(v) and q = u s(v) σ(v). By considering the shortest prefix with the same content, we obtain

t_{n}s(p) σ(p) = t_{n}s(q) σ(q).  (14)

580 M. Petrich; V. Silva: Relatively Free *-Bands

Writing r = u γ_{t_{m}}(v) w, we obtain

t_{n}(r) = t_{n}s(r) σ(r) \overline{t_{n−1}}(r)
= t_{n}s(p) σ(p) \overline{t_{n−1}}(r)  since ♯(uv) = k
= t_{n}s(q) σ(q) \overline{t_{n−1}}(r)  by (14)
= t_{n}s(uvw) σ(uvw) \overline{t_{n−1}}(r)  since ♯(uv) = k
= t_{n}s(uvw) σ(uvw) \overline{t_{n−1}}(uvw)  by (13)
= t_{n}(uvw). □

Lemma 4.3. For t ∈ {χ, i}, 3 ≤ n ≤ m and u, v, w ∈ F^{1}, we have

γ_{t_{n}}(u γ_{t_{m}}(v) w) = γ_{t_{n}}(uvw).

Proof. By definition of γ_{t_{n}}, we only need to show that

η(u γ_{t_{m}}(v) w) = η(uvw)

for η ∈ \{t_{n−1}s, σ, ε, \overline{t_{n−1}}e\}. Writing r = u γ_{t_{m}}(v) w, by Lemma 4.1 we have

\overline{r} = \overline{w} \overline{γ_{t_{m}}(v)} \overline{u} = \overline{w} γ_{t_{m}}(\overline{v}) \overline{u}.  (15)

We obtain

t_{n−1}s(r) = st_{n−1}(r)  by Lemma 3.2
= st_{n−1}(uvw)  by Lemma 4.2
= t_{n−1}s(uvw)  by Lemma 3.2,
σ(r) = σt_{n−1}(r)  by Lemma 3.2
= σt_{n−1}(uvw)  by Lemma 4.2
= σ(uvw)  by Lemma 3.2,
ε(r) = σ(\overline{r}) = σ(\overline{w} γ_{t_{m}}(\overline{v}) \overline{u})  by (15)
= σ(\overline{w} \overline{v} \overline{u})  by the above
= σ(\overline{uvw}) = ε(uvw),
\overline{t_{n−1}}e(r) = \overline{t_{n−1}(\overline{e(r)})} = \overline{t_{n−1}s(\overline{r})}
= \overline{t_{n−1}s(\overline{w} γ_{t_{m}}(\overline{v}) \overline{u})}  by (15)
= \overline{t_{n−1}s(\overline{w} \overline{v} \overline{u})}  by the above
= \overline{t_{n−1}s(\overline{uvw})} = \overline{t_{n−1}(\overline{e(uvw)})} = \overline{t_{n−1}}e(uvw). □

Lemma 4.4. For γ ∈ Γ and u, v ∈ F^{1}, we have γ(γ(u) γ(v)) = γ(uv).

M. Petrich; V. Silva: Relatively Free *-Bands 581

Proof. Apply Lemma 4.3 twice. □ Lemma 4.5. For γ ∈ Γ and w ∈ F^{1}, we have

γ(w^{2}) = γ(w) = γ(ww^{∗}w).

Proof. For γ = γ_{t_{n}} and w ∈ F, we have

\begin{aligned}γ_{t_{n}}(w^{2}) & = & t_{n−1}s(w^{2}) σ(w^{2}) ε(w^{2}) \overline{t_{n−1}}e(w^{2}) \\ & = & t_{n−1}s(w) σ(w) ε(w) \overline{t_{n−1}}e(w) = γ_{t_{n}}(w). \end{aligned}

The equality γ(w) = γ(ww^{∗}w) is proved similarly. □ Lemma 4.6. Let γ ∈ Γ, u, v ∈ F^{1} and φ ∈ End(F) be such that γ(u) = γ(v). Then γφ(u) = γφ(v). Proof. We need to show that ηφ(u) = ηφ(v) for η ∈ \{t_{n−1}s, σ, ε, \overline{t_{n−1}}e\}. Since t_{n−1}γ_{t_{n}} = t_{n−1} by Lemma 4.2, γ_{t_{n}}(u) = γ_{t_{n}}(v) yields that t_{n−1}(u) = t_{n−1}(v). Thus

t_{n−1}sφ(u) = st_{n−1}φ(u)  by Lemma 3.2
= st_{n−1}φ(v)  by Lemma 3.8
= t_{n−1}sφ(v)  by Lemma 3.2,
σφ(u) = σt_{n−1}φ(u)  by Lemma 3.2
= σt_{n−1}φ(v)  by Lemma 3.8
= σφ(v)  by Lemma 3.2.

By Lemma 4.1, we also have

γ_{t_{n}}(\overline{u}) = \overline{γ_{t_{n}}(u)} = \overline{γ_{t_{n}}(v)} = γ_{t_{n}}(\overline{v}).

Since \overline{φ} ∈ End(F) by Lemma 3.7, it follows that

εφ(u) = σ(\overline{φ(u)}) = σ\overline{φ}(\overline{u}) = σ\overline{φ}(\overline{v})  by the above
= σ(\overline{φ(v)}) = εφ(v),
\overline{t_{n−1}}eφ(u) = \overline{t_{n−1}(\overline{eφ(u)})} = \overline{t_{n−1}s(\overline{φ(u)})} = \overline{t_{n−1}s\overline{φ}(\overline{u})}
= \overline{t_{n−1}s\overline{φ}(\overline{v})}  by the above
= \overline{t_{n−1}s(\overline{φ(v)})} = \overline{t_{n−1}(\overline{eφ(v)})} = \overline{t_{n−1}}eφ(v). □

We now collect some important properties of the operator γ.

Lemma 4.7. Let γ ∈ Γ and denote by γ̂ the equivalence relation on F induced by γ, i.e. γ̂ is the kernel of γ. Then γ̂ is a fully invariant *-band congruence on F.

Proof. For any u, v, w, z ∈ F, we have

u γ̂ v, w γ̂ z ⇒ uw γ̂ vz  by Lemma 4.4
u γ̂ v ⇒ u^{∗} γ̂ v^{∗}  by Lemma 4.1
w γ̂ w^{2}  by Lemma 4.5
w γ̂ ww^{∗}w  by Lemma 4.5
u γ̂ v ⇒ φ(u) γ̂ φ(v)  by Lemma 4.6

for any *-endomorphism φ of F. □

We define a multiplication on γ(F) by u ⋆ v = γ(uv). In view of Lemma 4.1, the unary *- operation on F maps γ(F) into itself. Hence we may keep the unary operation of F restricted to γ(F). It now follows from Lemmas 4.4 and 4.1 that γ is a *-homomorphism of F onto γ(F) with modified operations, and thus γ(F) ≅ F/γ̂. By Lemma 4.7, γ̂ is a fully invariant *-band congruence and hence γ(F) is a V-free *-band on X for some *-band variety V. Our task in the next section is to identify the variety V by studying the identities in our diagram of the lattice of *-band varieties.

5. Identification of varieties

We have concluded the preceding section with the collection {γ̂ | γ ∈ Γ} of fully invariant *-band congruences on F. Our purpose now is to identify the variety V which corresponds to a given γ̂ in the usual antiisomorphism of the lattice of *-band varieties and fully invariant *-band congruences on F by means of our diagram of the former lattice. In this way, we shall include most of the diagram.

The following system of words was introduced in [4]:

G_{2} = x_{2}x_{1},  H_{2} = x_{2},  I_{2} = x_{2}x_{1}x_{2}

and for n > 2, defined inductively

G_{n} = x_{n}\overline{G_{n−1}},  T_{n} = G_{n}x_{n}\overline{T_{n−1}}  (T ∈ \{H, I\}).

We assume that t = h or t = χ if and only if T = H and t = i if and only if T = I. The lattice of all *-band varieties was determined in [1]. That the bases for these varieties

are as shown in the diagram was proved in [5].

[figure: [x = x]; [G_{4} = I_{4}]; [G_{4} = H_{4}]; [G_{3} = I_{3}]; [G_{3} = H_{3}]; [G_{2} = I_{2}]; [x = xyx]; [x = y]]

The lattice of *-band varieties

Lemma 5.1. For t ∈ {h, i} and m ≥ n ≥ 2, we have

t_{n}(G_{m}) = t_{n}(T_{m}), t_{n}(\overline{G_{m+1}}) = t_{n}(\overline{T_{m+1}}).

Proof. We use induction on n.

Let n = 2. The equality t_{2}(G_{m}) = t_{2}(T_{m}) follows from the definitions. Since \overline{G_{m+1}} = G_{m}x_{m+1} and \overline{T_{m+1}} = T_{m}x_{m+1}\overline{G_{m+1}}, we obtain the second equality from the first.

Now let n > 2 and assume that the lemma holds for all values smaller than n. Then we have

t_{n}s(T_{m}) = t_{n}s(G_{m}x_{m}\overline{T_{m−1}}) = t_{n}s(G_{m})  since c(G_{m}) = c(T_{m}),
σ(T_{m}) = σ(G_{m}x_{m}\overline{T_{m−1}}) = σ(G_{m})  since c(G_{m}) = c(T_{m}),
\overline{t_{n−1}}(T_{m}) = \overline{t_{n−1}(\overline{T_{m}})} = \overline{t_{n−1}(\overline{G_{m}})}  by induction on n
= \overline{t_{n−1}}(G_{m})

and so t_{n}(G_{m}) = t_{n}(T_{m}). Also

t_{n}s(\overline{T_{m+1}}) = t_{n}s(T_{m}x_{m+1}\overline{G_{m+1}}) = t_{n}(T_{m}) = t_{n}(G_{m})  by the above
= t_{n}s(G_{m}x_{m+1}) = t_{n}s(\overline{G_{m+1}}),
σ(\overline{T_{m+1}}) = σ(T_{m}x_{m+1}\overline{G_{m+1}}) = x_{k+1} = σ(G_{m}x_{m+1}) = σ(\overline{G_{m+1}}),
\overline{t_{n−1}}(\overline{T_{m+1}}) = \overline{t_{n−1}(T_{m+1})} = \overline{t_{n−1}(G_{m+1})}  by induction on n
= \overline{t_{n−1}}(\overline{G_{m+1}}),

584 M. Petrich; V. Silva: Relatively Free *-Bands

and hence t_{n}(\overline{G_{m+1}}) = t_{n}(\overline{T_{m+1}}). □ Lemma 5.2. For {p, q} = {χ, i} and n ≥ 2, we have p_{n}(G_{n}) ≠ p_{n}(Q_{n}). Proof. We use induction on n. Let n = 2. Since

\overline{χ_{1}}(I_{2}) = x_{2} ≠ x_{1} = \overline{χ_{1}}(G_{2}),

it follows that χ_{2}(I_{2}) ≠ χ_{2}(G_{2}). Clearly, i_{2}(H_{2}) = x_{2} ≠ x_{2}x_{1} = i_{2}(G_{2}).

Now let n > 2 and assume that the lemma holds for n − 1. We consider first the case p = i and n = 3. If i_{3}(H_{3}) = i_{3}(G_{3}), then

i_{2}(\overline{H_{3}}) = i_{2}\overline{i_{3}}(\overline{H_{3}})  by Lemma 3.6
= i_{2}(\overline{i_{3}(H_{3})}) = i_{2}(\overline{i_{3}(G_{3})}) = i_{2}(\overline{G_{3}})  by symmetry.

However,

i_{2}(\overline{H_{3}}) = i_{2}(x_{2}x_{3}x_{2}x_{1}x_{3}) = x_{2}x_{3}x_{1} ≠ x_{2}x_{1}x_{3} = i_{2}(x_{2}x_{1}x_{3}) = i_{2}(\overline{G_{3}})

and so i_{3}(H_{3}) ≠ i_{3}(G_{3}).

Assume now that p = χ or n > 3. If p_{n}(Q_{n}) = p_{n}(G_{n}), then

p_{n−1}(Q_{n−1}) = p_{n−1}s(Q_{n−1}x_{n}\overline{G_{n}}) = p_{n−1}s(\overline{Q_{n}})
= sp_{n−1}(\overline{Q_{n}})  by Lemma 3.2
= sp_{n−1}\overline{p_{n}}(\overline{Q_{n}})  by Lemma 3.6
= sp_{n−1}(\overline{p_{n}(Q_{n})}) = sp_{n−1}(\overline{p_{n}(G_{n})})
= p_{n−1}s(\overline{G_{n}})  by symmetry
= p_{n−1}s(G_{n−1}x_{n}) = p_{n−1}(G_{n−1}),

contradicting the induction hypothesis. Therefore p_{n}(Q_{n}) ≠ p_{n}(G_{n}). □ Lemma 5.3. Let n ≥ 3.

(i) γ_{χ_{n}}(G_{n+1}) = γ_{χ_{n}}(H_{n+1}), γ_{χ_{n}}(\overline{G_{n}}) ≠ γ_{χ_{n}}(\overline{I_{n}}). (ii) γ_{i_{n}}(G_{n}) = γ_{i_{n}}(I_{n}), γ_{i_{n}}(\overline{G_{n}}) ≠ γ_{i_{n}}(\overline{H_{n}}). Proof. (i) We show that η(H_{n+1}) = η(G_{n+1}) for η ∈ \{χ_{n−1}s, σ, ε, \overline{χ_{n−1}}e\}. Indeed,

χ_{n−1}s(H_{n+1}) = χ_{n−1}s(G_{n+1}x_{n+1}\overline{H_{n}}) = χ_{n−1}s(G_{n+1}),
σ(H_{n+1}) = σ(G_{n+1}x_{n+1}\overline{H_{n}}) = σ(G_{n+1}),
ε(H_{n+1}) = ε(G_{n+1}x_{n+1}\overline{H_{n}}) = x_{n+1} = ε(x_{n+1}\overline{G_{n}}) = ε(G_{n+1}),
\overline{χ_{n−1}}e(H_{n+1}) = \overline{χ_{n−1}}e(G_{n+1}x_{n+1}\overline{H_{n}}) = \overline{χ_{n−1}}(\overline{H_{n}}) = \overline{χ_{n−1}(H_{n})}
= \overline{χ_{n−1}(G_{n})}  by Lemma 5.1
= \overline{χ_{n−1}}(\overline{G_{n}}) = \overline{χ_{n−1}}e(x_{n+1}\overline{G_{n}}) = \overline{χ_{n−1}}e(G_{n+1}).

Now suppose that γ_{χ_{n}}(\overline{G_{n}}) = γ_{χ_{n}}(\overline{I_{n}}) for some n ≥ 3. Then we would have
χ_{n−1}(I_{n−1}) = χ_{n−1}s(I_{n−1}x_{n}\overline{G_{n}}) = χ_{n−1}s(\overline{I_{n}}) = sγ_{χ_{n}}(\overline{I_{n}})
= s(\overline{γ_{χ_{n}}(I_{n})})  by Lemma 4.1
= s(\overline{γ_{χ_{n}}(G_{n})})  by hypothesis
= χ_{n−1}s(\overline{G_{n}})  by symmetry
= χ_{n−1}s(G_{n−1}x_{n}) = χ_{n−1}(G_{n−1}),

contradicting Lemma 5.2. Therefore γ_{χ_{n}}(\overline{G_{n}}) ≠ γ_{χ_{n}}(\overline{I_{n}}) for every n ≥ 3.

(ii) Similar. □

6. Main result

In order to collect all the information we need, we first introduce some notation. Let X be a nonempty set.

Let RB = [x = xyx]. Define a mapping ψ_{RB} on F by

ψ_{RB} : w → h_{2}(w) \overline{h_{2}}(w).

On the set ψ_{RB}(F) define a multiplication by xy ⋆ wz = xz and a unary operation by (xy)^{∗} = y^{∗}x^{∗}.

Let S = [G_{2} = I_{2}]. Assume that X is totally ordered and define a mapping ψ_{S} on F by

ψ_{S} : w → x_{1}x_{2} … x_{n}

where c(w) = {x_{1}, x_{2}, …, x_{n}} and x_{1} < x_{2} < … < x_{n}. On the set ψ_{S}(F) define a multiplication by

x_{1}x_{2} … x_{m} ⋆ y_{1}y_{2} … y_{n} = z_{1}z_{2} … z_{k}

where

{x_{1}, x_{2}, …, x_{m}, y_{1}, y_{2}, …, y_{n}} = {z_{1}, z_{2}, …, z_{k}}, z_{1} < z_{2} < … < z_{k}  (16)

and identity mapping as unary operation.

Let N B = [G_{3} = H_{3}]. Define a mapping ψ_{N B} on F by

ψ_{N B} : w → h_{2}(w) ψ_{S}(w) \overline{h_{2}}(w).

On the set ψ_{N B}(F) define a multiplication by

ix_{1}x_{2} … x_{m}j ⋆ py_{1}y_{2} … y_{n}q = iz_{1}z_{2} … z_{k}q

in the notation (16) and a unary operation by

(ix_{1}x_{2} … x_{m}j)^{∗} = j^{∗}x_{1}x_{2} … x_{m}i^{∗}.

For γ ∈ Γ, say γ = γ_{t_{n}}, t ∈ {h, i}, let ψ_{[G_{n}=T_{n}]} = γ. On the set γ(F) define a multiplication by u ⋆ v = γ(uv) and consider the unary operation on F restricted to γ(F).

Define an operator b on F inductively on ♯(w) by

b(w) = bs(w) σ(w) [bs(w^{∗}) σ(w^{∗})]^{∗}  (17)

(this is taken from [3], Section 4, where b, s and σ are denoted by b^{∗}, s_{X} and σ_{X}, respectively). Let B be the variety of *-bands. Define a mapping ψ_{B} on F by

ψ_{B} : w → b(w).

On the set b(F) define a multiplication by u ⋆ v = b(uv) and consider the unary operation on F restricted to b(F).

We are finally ready for the desired result.

Theorem 6.1. Let X be a nonempty set and V be a nontrivial *-band variety. Then ψ_{V} is a *-homomorphism of F onto ψ_{V}(F) which induces the least V-congruence on F. Therefore ψ_{V}(F) is a V-free *-band on X.

Proof. The argument for the varieties RB, S and N B is straightforward and is omitted.

Let γ = γ_{h_{n}} with n ≥ 4 and V = [G_{n} = H_{n}]. The remarks at the end of the preceding section show that γ is a *-homomorphism of *-semigroups which induces the least W-congruence on F for some *-band variety W. By Lemma 5.3(i), we have γ(G_{n}) = γ(H_{n}). By Lemma 4.6, it follows that γφ(G_{n}) = γφ(H_{n}) for every φ ∈ End(F). Thus γ(F) satisfies the identity G_{n} = H_{n} and so γ(F) ∈ V. Therefore W ⊆ V.

Since [G_{n−1} = I_{n−1}] is the variety lying immediately below V in the lattice of varieties of *-bands and γ(G_{n−1}) ≠ γ(I_{n−1}) by Lemma 5.3(i), it follows that γ(F) ∉ [G_{n−1} = I_{n−1}] and so W ⊈ [G_{n−1} = I_{n−1}]. Thus W = V.

The case γ = γ_{i_{n}} is similar.

The case of B is essentially the content of ([3], Theorem 4.5). □

We now establish a few properties of the mappings ψ_{V}. Let Ω be the set of nontrivial *-band varieties different from S and N B. Note that Ω is a chain with least element RB and greatest element B. The remaining elements are the varieties

H_{n} = [G_{n} = H_{n}] for n ≥ 4, I_{n} = [G_{n} = I_{n}] for n ≥ 3

with the order

H_{m} ⊆ H_{n} ⇔ I_{m} ⊆ I_{n} ⇔ m ≤ n, I_{m} ⊂ H_{n} ⇔ m < n,

and ψ_{H_{n}} = γ_{h_{n}}, ψ_{I_{n}} = γ_{i_{n}}.

Our first result concerns properties of individual ψ_{V}.

Proposition 6.2. Let V ∈ Ω and set ψ = ψ_{V}. Then ψ = \overline{ψ} = ψ^{∗} = ψ^{2}.

Proof. The verification for V = RB is straightforward and is omitted. The case of V different from RB and B was handled in Lemma 4.1 in terms of γ ∈ Γ. For V = B, we have to prove the corresponding statements for b. For \overline{b} = b, the proof of ([7], Lemma 6.1) remains valid for *-bands. An obvious variant of this proof will show that also b^{∗} = b holds. For b^{2} = b, the proof of ([3], Lemma 2.2(iii)) remains valid in the case of *-bands as well. □

For V ∈ \{S, N B\}, we have ψ_{V}^{2} = ψ_{V} but \overline{ψ_{V}} ≠ ψ_{V} and ψ_{V}^{∗} ≠ ψ_{V}. This is not surprising in view of the nonintrinsic nature of the definition of ψ_{S} and ψ_{N B}, for it depends on the choice of the total order on X. When ψ_{V}^{2} = ψ_{V}, we have that ψ_{V}(F) coincides with the set of fixed points of ψ_{V}.

The form of the inductive formula (17) is quite different from that of any γ in Γ. We can modify that formula by observing that

b(w) = bs(w) σ(w) [bs(w^{∗}) σ(w^{∗})]^{∗}
= bs(w) σ(w) (σ(w^{∗}))^{∗} (bs(w^{∗}))^{∗} = bs(w) σ(w) σ^{∗}(w) bs^{∗}(w)
= bs(w) σ(w) ε(w) be(w)  by Lemma 2.1

which is closer to the definition of γ in Γ. We can get even closer by observing that \overline{b} = b, for then

b(w) = bs(w) σ(w) ε(w) \overline{b}e(w).

Next we compare two *-band varieties U and V and the corresponding ψ_{U} and ψ_{V}.

Proposition 6.3. Let U, V be *-band varieties. Then U ⊆ V if and only if ψ_{U}ψ_{V} = ψ_{U}.

Proof. For a variety V of *-bands, let ∼_{V} denote the least V-congruence on F. By Theorem 6.1, ∼_{V} is induced by ψ_{V}. Let U, V be *-band varieties.

Assume that U ⊆ V. Then ∼_{V} ⊆ ∼_{U}. Let w ∈ F. By Proposition 6.2, we have ψ_{V}ψ_{V}(w) = ψ_{V}(w) and so ψ_{V}(w) ∼_{V} w. Since ∼_{V} ⊆ ∼_{U}, it follows that ψ_{V}(w) ∼_{U} w and so ψ_{U}ψ_{V}(w) = ψ_{U}(w). Hence ψ_{U}ψ_{V} = ψ_{U}.

Conversely, assume that ψ_{U}ψ_{V} = ψ_{U}. Let w, z ∈ F be such that w ∼_{V} z. Then ψ_{V}(w) = ψ_{V}(z) and so

ψ_{U}(w) = ψ_{U}ψ_{V}(w) = ψ_{U}ψ_{V}(z) = ψ_{U}(z).

Hence w ∼_{U} z and so ∼_{V} ⊆ ∼_{U}. Thus U ⊆ V. □

References

[1] Adair, C. L.: Bands with involution. J. Algebra 75 (1982), 297–314.

[2] Fennemore, C. F.: All varieties of bands I, II. Math. Nachr. 48 (1971), 237–252, 253–262.

[3] Gerhard J. A.; Petrich, M.: Free bands and free *-bands. Glasgow Math. J. 28 (1986), 161–179.

[4] Gerhard, J. A.; Petrich, M.: Varieties of bands revisited. Proc. London Math. Soc. (3) 58 (1989), 323–350.

[5] Petrich, M.: Identities without the star for *-bands. Algebra Universalis 36 (1996), 46–65.

[6] Petrich, M.; Silva, P. V.: Relatively free bands. Preprint, Centro de Matemática do Porto.

[7] Petrich, M.; Silva, P. V.: Structure of relatively free bands. Preprint, Centro de Matemática do Porto.

[8] Yamada, M.: Finitely generated free *-bands. Semigroup Forum 29 (1984), 13–16.

Received September 15, 1999