## IS:P

Solutions to exercises from Introduction to Semigroups by Petrich.

Chapter 1: Preliminaries
Chapter 2: Semilattice Decompositions
Chapter 3: Ideal Extensions
Chapter 4: Completely Regular Semigroups
Chapter 5: The Translational Hull

## Solutions

• Chapter 1: Preliminaries
(I.3.3) (I.3.5) (I.3.7) (I.3.9) (I.4.2) (I.4.5) (I.4.13) (I.5.5) (I.5.6) (I.5.8) (I.5.12) (I.5.14) (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26)
• Chapter 2: Semilattice Decompositions
§2.1: Subdirect Products
(II.1.5) (II.1.6) (II.1.7) (#1) (#2) (#3) (#4) (#5) (#6)
§2.2: Completely Prime Ideals and Filters
(II.2.3) (II.2.15) (II.2.17) (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8)
§2.3: Completely Semiprime Ideals and $\mathfrak{N}$-Subsets
(II.3.9) (II.3.13) (II.3.14) (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9)
§2.4: Semilattices of Simple Semigroups
(II.4.4) (II.4.8) (II.4.10) (II.4.11) (II.4.12) (#1) (#2) (#3) (#4) (#5) (#6) (#7)
§2.5: Weakly Commutative Semigroups
(II.5.4) (#1) (#2) (#3) (#4) (#5) (#6)
§2.6: Separative Semigroups
(II.6.5) (II.6.7) (#1) (#2) (#3) (#4)
§2.7: $\mathfrak{N}$-Semigroups
(II.7.6) (#1) (#2) (#3) (#4) (#5) (#6) (#7)
• Chapter 3: Ideal Extensions
§3.1: Extensions and Translations
(III.1.7) (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9)
§3.2: Extensions of a Weakly Reductive Semigroup
(#1) (#2) (#3) (#4) (#5)
§3.3: Strict and Pure Extensions
(#1) (#2) (#3)
§3.4: Retract Extensions
(III.4.2) (III.4.6) (#1) (#2) (#3) (#4) (#5) (#6) (#7)
§3.5: Dense Extensions
(III.5.3) (III.5.10) (III.5.18) (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8)
§3.6: Extensions of an Arbitrary Semigroup
(III.6.4) (III.6.11) (#1) (#2) (#3) (#4)
§3.7: Semilattice Compositions
(III.7.7) (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11)
• Chapter 4: Completely Regular Semigroups
§4.1: Generalities
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10)
§4.2: Completely Simple Semigroups
(IV.2.9) (#1) (#2) (#3) (#4)
§4.3: Semilattices of Rectangular Groups
(IV.3.5) (IV.3.6) (IV.3.11) (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9)
§4.4: Strong Semilattices of Completely Simple Semigroups
(IV.4.7) (IV.4.8) (#1) (#2) (#3) (#4) (#5) (#6) (#7)
§4.5: Subdirect Products of a Semilattice and a Completely Simple Semigroup
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14)
• Chapter 5: The Translational Hull
§5.1: General Properties
(V.1.2) (V.1.4) (V.1.8) (V.1.15) (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13)
§5.2: Isomorphisms and Automorphisms
(#1) (#2) (#3)
§5.3: Rees Matrix Semigroups
(V.3.15) (#1) (#2) (#3) (#4) (#5) (#6)
§5.4: Inverse Semigroups
(V.4.2) (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12)
§5.5: Brandt Semigroups
(V.5.5) (#1) (#2) (#3) (#4) (#5)
§5.6: Semilattices of Groups
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9)