Given a semilattice X we study the algebraic properties of the semigroup υ(X) of upfamilies on X.The semigroup υ(X) contains the Stone-Čech extension β(X), the superextension λ(X), and the space of filters ϕ(X) on X as closed subsemigroups. We prove that υ(X) is a semilattice iff λ(X) is a semilattice iff ϕ(X) is a semilattice iff the semilattice X is finite and linearly ordered. We prove that the semigroup β(X) is a band if and only if X has no infinite antichains, and the semigroup λ(X) is commutative if and only if X is a bush with finite branches. IntroductionOne of powerful tools in the modern Combinatorics of Numbers is the method of ultrafilters based on the fact that each (associative) binary operation * : X × X → X defined on a discrete topological space X extends to a right-topological (associative) operation * : [11]. The Stone-Čech extension β(X) is the space of ultrafilters on X. The extension of the operation from X to β(X) can be defined by the simple formula:Endowed with the so-extended operation, the Stone-Čech compactification β(X) becomes a compact righttopological semigroup. The algebraic properties of this semigroup (for example, the existence of idempotents or minimal left ideals) have important consequences in combinatorics of numbers, see [9], [11].In [8] it was observed that the binary operation * extends not only to β(X) but also to the space υ(X) of all upfamilies on X. By definition, a family F of non-empty subsets of a discrete space X is called an upfamily if for any sets A ⊂ B ⊂ X the inclusion A ∈ F implies B ∈ F . The space υ(X) is a closed subspace of the double power-set P(P(X)) endowed with the compact Hausdorff topology of the Tychonoff power {0, 1} P(X) . In the papers [7], [8], [1]-[4] the space υ(X) was denoted by G(X) and its elements were called inclusion hyperspaces 1 .The extension of a binary operation * from X to υ(X) can be defined in the same way as for ultrafilters, i.e., by the formula (1) applied to any two upfamilies U, V ∈ υ(X). If X is a semigroup, then υ(X) is a compact Hausdorff right-topological semigroup containing β(X) as closed subsemigroups. The algebraic properties of this semigroups were studied in details in [8].The space υ(X) of upfamilies over a discrete space X contains many interesting subspaces. First we recall some definitions. An upfamily A ∈ υ(X) is defined to be • a filter if A 1 ∩ A 2 ∈ A for all sets A 1 , A 2 ∈ A;• an ultrafilter if A = A ′ for any filter A ′ ∈ υ(X) containing A;• linked if A ∩ B = ∅ for any sets A, B ∈ A;• maximal linked if A = A ′ for any linked upfamily A ′ ∈ υ(X) containing A.1991 Mathematics Subject Classification. 06A12, 20M10. Key words and phrases. semilattice, band, commutative semigroup, the space of upfamilies, the space of filters, the space of maximal linked systems, superextension. 1 We decided to change the terminology and notation after discovering the paper [12, 2.7.4] that discusses monadic properties of the up-set functor υ.

A subset B of a group G is called a difference basis of G if each element g ∈ G can be written as the difference g = ab −1 of some elements a, b ∈ B. The smallest cardinality |B| of a difference basis B ⊂ G is called the difference size of G and is denoted by ∆ [G]. The fraction ð[G] := ∆[G]/ |G| is called the difference characteristic of G. We prove that for every n ∈ N the dihedral group D 2n of order 2n has the difference characteristic√ 586 ≈ 1.983. Moreover, if n ≥ 2 · 10 15 , then ð[D 2n ] < 4 √ 6 ≈ 1.633. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality ≤ 80. 1991 Mathematics Subject Classification. 05B10, 05E15, 20D60.

A family $\mathcal{A}$ of non-empty subsets of a set $X$ is called an upfamily if for each set $A\in\mathcal{A}$ any set $B\supset A$ belongs to $\mathcal{A}$. An upfamily $\mathcal L$ of subsets of $X$ is said to be linked if $A\cap B\ne\emptyset$ for all $A,B\in\mathcal L$. A linked upfamily $\mathcal M$ of subsets of $X$ is maximal linked if $\mathcal M$ coincides with each linked upfamily $\mathcal L$ on $X$ that contains $\mathcal M$. The superextension $\lambda(X)$ consists of all maximal linked upfamilies on $X$. Any associative binary operation $* : X\times X \to X$ can be extended to an associative binary operation $\circ: \lambda(X)\times\lambda(X)\to\lambda(X)$ by the formula $\mathcal L\circ\mathcal M=\Big\langle\bigcup_{a\in L}a*M_a:L\in\mathcal L,\;\{M_a\}_{a\in L}\subset\mathcal M\Big\rangle$ for maximal linked upfamilies $\mathcal{L}, \mathcal{M}\in\lambda(X)$. In the paper we describe superextensions of all three-element semigroups up to isomorphism.

Given a cyclic semigroup S we study right and left zeros, singleton left ideals, the minimal ideal, left cancelable and right cancelable elements of superextensions λ(S) and characterize cyclic semigroups whose superextensions are commutative.

A family A of non-empty subsets of a set X is called an upfamily if for each set A ∈ A any set B ⊃ A belongs to A. An upfamily L of subsets of X is said to be linked if A ∩ B ̸ = ∅ for all A, B ∈ L. A linked upfamily M of subsets of X is maximal linked if M coincides with each linked upfamily L on X that contains M. The superextension λ(X) consists of all maximal linked upfamilies on X. Any associative binary operation * : X ×X → X can be extended to an associative binary operation * : λ(X) × λ(X) → λ(X). In the paper we study automorphisms of superextensions of semigroups and describe the automorphism groups of superextensions of null semigroups, almost null semigroups, right zero semigroups and left zero semigroups. Also we find the automorphism groups of superextensions of all semigroups S of order |S| ≤ 3.