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Stockholders' equity - Changes in Common Stock and Treasury Stock (Details) - shares shares in Millions | 12 Months Ended
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Dec. 31, 2018 | Dec. 31, 2017 | Dec. 31, 2016
Common Stock |  |  |
Increase (Decrease) in Stockholders' Equity [Roll Forward] |  |  |
Beginning Balance (in shares) | 53.4 | 52.8 | 52.6
Stock-based compensation (in shares) | 0.8 | 0.6 | 0.2
Ending Balance (in shares) | 54.2 | 53.4 | 52.8
Treasury Stock |  |  |
Increase (Decrease) in Stockholders' Equity [Roll Forward] |  |  |
Beginning Balance (in shares) | 2.9 | 2.7 | 2.7
Stock-based compensation (in shares) | 0.2 | 0.2 | 0.0
Ending Balance (in shares) | 3.1 | 2.9 | 2.7

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- Definition A roll forward is a reconciliation of a concept from the beginning of a period to the end of a period. + References No definition available. + Details
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- Definition Number of shares (or other type of equity) issued during the period as a result of any equity-based compensation plan other than an employee stock ownership plan (ESOP), net of any shares forfeited. Shares issued could result from the issuance of restricted stock, the exercise of stock options, stock issued under employee stock purchase plans, and/or other employee benefit plans. + References Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef -Publisher FASB -Name Accounting Standards Codification -Topic 505 -SubTopic 10 -Section 50 -Paragraph 2 -URI http://asc.fasb.org/extlink&oid=109259400&loc=d3e21463-112644 Reference 2: http://fasb.org/us-gaap/role/ref/legacyRef -Publisher FASB -Name Accounting Standards Codification -Topic 505 -SubTopic 10 -Section S99 -Paragraph 1 -Subparagraph (SX 210.3-04) -URI http://asc.fasb.org/extlink&oid=27012166&loc=d3e187085-122770 Reference 3: http://fasb.org/us-gaap/role/ref/legacyRef -Publisher FASB -Name Accounting Standards Codification -Topic 210 -SubTopic 10 -Section S99 -Paragraph 1 -Subparagraph (SX 210.5-02(28)) -URI http://asc.fasb.org/extlink&oid=6877327&loc=d3e13212-122682 Reference 4: http://fasb.org/us-gaap/role/ref/legacyRef -Publisher FASB -Name Accounting Standards Codification -Topic 210 -SubTopic 10 -Section S99 -Paragraph 1 -Subparagraph (SX 210.5-02(29)) -URI http://asc.fasb.org/extlink&oid=6877327&loc=d3e13212-122682 + Details
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