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The final publication is available at Springer via http://dx.doi.org/10.1007/s00224-017-9803-8 ; The state complexity of a regular language is the number of states in a minimal deterministic finite automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that nn−1, nn−1 + n − 1, and nn−2 + (n − 2)2n−2 + 1 are tight upper bounds on the syntactic complexities of right ideals and prefix-closed languages, left ideals and suffix-closed languages, and two-sided ideals and factor-closed languages, respectively. Moreover, we show that the transition semigroups meeting the upper bounds for all three types of ideals are unique, and the numbers of generators (4, 5, and 6, respectively) cannot be reduced. ; Natural Sciences and Engineering Research Council of Canada (NSERC) grant No. OGP000087 ; National Science Centre, Poland project number 2013/09/N/ST6/01194 ; NSERC Postgraduate Scholarship and a Graduate Award from the Department of Computer Science ; co-financed by the European Union under the European Social Fund's project "International computer science and applied mathematics for business study programme at the University of Wrocław